Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Let Xλ and X′λ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], whe...

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1. Verfasser:	Kloosterman, R.
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spelling	irk-123456789-1492752019-02-20T01:23:59Z Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces Kloosterman, R. Let Xλ and X′λ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249]. 2017 Article Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces / R. Kloosterman // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 21 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14G10; 11G25; 14C22; 14J28; 14J70; 14Q10 DOI:10.3842/SIGMA.2017.087 http://dspace.nbuv.gov.ua/handle/123456789/149275 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let Xλ and X′λ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].
format	Article
author	Kloosterman, R.
spellingShingle	Kloosterman, R. Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kloosterman, R.
author_sort	Kloosterman, R.
title	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
title_short	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
title_full	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
title_fullStr	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
title_full_unstemmed	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces
title_sort	zeta functions of monomial deformations of delsarte hypersurfaces
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/149275
citation_txt	Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces / R. Kloosterman // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 21 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT kloostermanr zetafunctionsofmonomialdeformationsofdelsartehypersurfaces
first_indexed	2025-07-12T21:13:44Z
last_indexed	2025-07-12T21:13:44Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 087, 22 pages Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces Remke KLOOSTERMAN Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste 63, 35121 Padova, Italy E-mail: klooster@math.unipd.it URL: http://www.math.unipd.it/~klooster/ Received June 09, 2017, in final form November 01, 2017; Published online November 07, 2017 https://doi.org/10.3842/SIGMA.2017.087 Abstract. Let Xλ and X ′ λ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi–Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249]. Key words: monomial deformation of Delsarte surfaces; zeta functions 2010 Mathematics Subject Classification: 14G10; 11G25; 14C22; 14J28; 14J70; 14Q10 1 Introduction Fix a finite field Fq and a positive integer n. In this paper we study a particular class of defor- mations of Delsarte hypersurfaces in Pn Fq . There has been an extensive study of the behaviour of the zeta function in families of varieties. First results were obtained by Dwork (e.g., [9]) and Katz [12]. In the latter paper the author studies a pencil of hypersurface in Pn and describe a differential equation, whose solution is the Frobenius matrix on the middle cohomology for a general member of this pencil. More recently, the behaviour of the zeta function acquired renewed interest because of two interesting (and very different) applications. Candelas, de la Ossa and Rodriguez–Villegas [6] studied the behaviour of the zeta family in a particular family of quintic threefolds in P4, with a particular interest in phenomena, analogous to phenomena occurring in characteristic zero related with mirror symmetry and let to many subsequent papers by various authors. Another application of Katz’ differential equation can be found in algorithms to determine the zeta function of a hypersurface efficiently (see [17, 18]). The main aim of this paper is to generalize and to comment on a recent result of Doran, Kelly, Salerno, Sperber, Voight and Whitcher [8] on the zeta function of certain pencils of Calabi–Yau hypersurfaces. For a more extensive discussion on the history of this particular result we refer to the introduction of [8]. To describe the main results from [8], fix a matrix A := (ai,j)0≤i,j≤n with nonnegative integral coefficients and nonzero determinant. Then with A we can associate the polynomial FA := n∑ i=0 n∏ j=0 x ai,j i . This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html mailto:klooster@math.unipd.it http://www.math.unipd.it/~klooster/ https://doi.org/10.3842/SIGMA.2017.087 http://www.emis.de/journals/SIGMA/modular-forms.html 2 R. Kloosterman Assume that the entries of A−1(1, . . . , 1)T are all positive, say 1 e (w0, . . . , wn) with e, wi ∈ Z>0. Then FA defines a hypersurface of degree e in PFq(w0, . . . , wn). Assume that we choose A such that gcd(q, e) = 1. (Equivalently, we may assume that gcd(det(A), q) = 1.) If the hypersurface is geometrically irreducible then we call it a Delsarte hypersurface. A sub- variety X ⊂ P(w0, . . . , wn) is called quasismooth if the affine quasicone of X is smooth away from the vertex. If FA defines a quasismooth hypersurface then FA is called an invertible poly- nomial. If FA = 0 defines a Calabi–Yau manifold, i.e., e = n + 1, then we can consider the one-parameter family XA,ψ given by the vanishing of FA − (n+ 1)ψ n∏ i=0 xi. The factor −(n+1) is included for historic reasons. In the sequel we will work with the parameter λ = −(n+ 1)ψ for simplicity. In a recent preprint Doran, Kelly, Salerno, Sperber, Voight and Whitcher [8] showed the following result (using Dwork cohomology and some results on the Picard–Fuchs equation): Theorem 1.1 ([8]). Let A and A′ be (n+ 1)× (n+ 1)-matrices with nonnegative entries such that FA and FA′ are invertible Calabi–Yau polynomials of degree n+ 1. Assume that (1, . . . , 1)T is an eigenvector of both A and A′ and that gcd(q, (n+ 1) det(A) det(A′)) = 1. Moreover, assume that (1, . . . , 1)A−1 and (1, . . . , 1)A′−1 are proportional. Then for any ψ ∈ Fq such that XA,ψ and XA′,ψ are smooth and nondegenerate we have that the polynomials( Z(XA,ψ, T ) n−1∏ i=0 ( 1− qiT ))(−1)n and ( Z(XA′,ψ, T ) n−1∏ i=0 ( 1− qiT ))(−1)n have a common factor of degree at least the order of the Picard–Fuchs equation of XA,ψ. For a precise definition of nondegenerate we refer to the paper [8]. The condition (1, 1, . . . , 1)T is an eigenvector of A implies that XA,ψ ⊂ Pn. The condition (1, . . . , 1)A−1 is proportional to (1, . . . , 1)A′−1 is the same as the condition dual weights being equal from the paper [8], whenever the latter condition is defined. In this paper we prove a generalisation of this result. We aim to allow more matrices A, more vectors a, to drop the Calabi–Yau assumption, to have a simpler nondegenerate assumption and to find a common factor of higher degree. Moreover, as a by-prodcut of our approach we obtain additional information on the degree of the factor found in [8]. To be more precise, we start again with an invertible (n+1)×(n+1)-matrix A such that XA,0 is quasismooth, but we drop the Calabi–Yau condition. Let d be an integer such that B := dA−1 has integral entries. Let w = (w0, . . . , wn) := B(1, . . . , 1)T. If all the wi are positive then FA defines a hypersurface in the weighted projective space P(w). Fix now a vector a := (a0, . . . , an) such that ai ∈ Z>0, the entries of b := aB are nonnegative and n∑ i=0 aiwi = d. Then FA,ψ := FA − (n + 1)ψ n∏ i=0 xaii defines a family of hypersurfaces Xψ in P(w) each birational to a quotient of Yψ ⊂ Pn given by n∑ i=0 ydi − (n+ 1)ψ n∏ i=0 ybii . Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 3 This is a one-dimensional monomial deformation of a Fermat hypersurface. It is easy to de- termine for which values of ψ the hypersurface is smooth [15, Lemma 3.7]. The idea to study Delsarte hypersurface by using their Fermat cover dates back to Shioda [20] and has then been used by many authors for to discuss solve problems concerning Delsarte hypersurfaces by consi- dering a similar problem on Fermat surfaces. Recent applications of this idea, in contexts similar to our setup, can be found in [4, 5, 13]. Take now a further (n+ 1)× (n+ 1) matrix A′ and a vector a′ yielding a second family X ′ψ in a possibly different weighted projective space. It is straightforward to show that if aA−1 and a′(A′)−1 are proportional then the fami- lies Xψ, X ′ψ have a common cover of the type Yψ, i.e., there exist subgroup schemes G and G′ of the scheme of automorphisms Aut(Yψ) such that G and G′ are defined over Fq, Yψ/G is birational to Xψ and Yψ/G ′ is birational to X ′ψ. The automorphisms in G and G′ are so-called torus or diagonal automorphisms, i.e., each automorphism multiplies a coordinate with a root of unity. In particular, G and G′ are finite abelian groups. We will use this observation to show that: Theorem 1.2. Let A and A′ be (n+ 1)× (n+ 1)-matrices with nonnegative entries, such that the entries of (w0, . . . , wn)T := A−1(1, . . . , 1)T and of (w′0, . . . , w ′ n)T := A′−1(1, . . . , 1)T are all positive and gcd(q,det(A) det(A′)) = 1. Fix two vectors a := (a0, . . . , an) and a′ := (a′0, . . . , a ′ n) consisting of nonnegative integers such that the equalities n∑ i=0 aiwi = 1 and n∑ i=0 a′iw ′ i = 1 hold and such that aA and a′A′ are proportional. Let Xψ, X ′ψ, Yψ, G and G′ as above. Denote with G.G′ the subgroup of Aut(Yψ) generated by G and G′. 1. If Yψ is smooth then the characteristic polynomial of Frobenius acting on Hn−1(Yψ)G.G ′ divides both the characteristic polynomial of Frobenius acting on Hn−1(Xψ) and the cha- racteristic polynomial of Frobenius on Hn−1(X ′ψ). 2. If, moreover, (1, . . . , 1) is an eigenvector of both A and A′ and both Xψ and X ′ψ are smooth then we have that the polynomials( Z(Xψ, T ) n−1∏ i=0 ( 1− qiT ))(−1)n and ( Z(X ′ψ, T ) n−1∏ i=0 ( 1− qiT ))(−1)n have a common factor of positive degree. In the second section we will prove this result under slightly weaker, but more technical hypothesis, see Theorem 2.18 and Corollary 2.21. Moreover, in Proposition 2.24 we will show that the factor constructed in the proof of Theorem 1.1 divides the characteristic polynomial of Frobenius acting on Hn−1(Yψ)G.G ′ . We will give examples where our factor has higher degree. Note that the quotient map Yψ 99K Xψ is a rational map. If it were a morphism then it is straightforward to show that the characteristic polynomial of Hn−1(Yψ)G.G ′ divides the characteristic polynomial of Frobenius on Hn−1(Xψ). Hence, large part of the proof is dedicated to show that passing to the open where the rational map is a morphism does not kill any part of Hn−1(Yψ)G.G ′ . In the course of the proof of Theorem 1.2 we show that we can decompose Hn−1(Xψ) as a direct sum of two Frobenius stable subspaces, namely Hn−1(Xψ) = Hn−1(Yψ)G ⊕ C. Similarly, we show that can decompose Hn−1(Yψ)G = Hn−1(Yψ)Gmax ⊕Wψ, where Wψ is Frobe- nius stable, and Gmax is the maximal group of torus automorphisms acting on the family Yψ. 4 R. Kloosterman The appearance of C is related with the fact that the quotient map is only a rational map rather than a morphism. For most choices of (a0, . . . , an) we have that C is independent of ψ and in that case we can express C in terms of the cohomology of cones over Fermat hypersurfaces. Hence the Frobenius action on C is easy to determine. To calculate the Frobenius action on the complementary subspace Hn−1(Yψ)G we can use the methods from [15] to express the zeta function in terms of generalised p-adic hypergeometric functions. This brings us to another observation from [8]: In [8, Section 5] the authors consider five families of quartic K3 surfaces which have a single common factor of the zeta function of degree 3. They show that every other zero of the characteristic polynomial of Frobenius on H2 is of the form q times a root of unity. Assuming the Tate conjecture for K3 surfaces (which is proven for most K3 surfaces anyway) we deduce that the (geometric) Picard number is at least 19. This result is a special case of the following phenomena: if for a lift to characteristic zero hn−1,0 ( Hn(Yψ)Gmax ) = hn−1,0 ( Hn−1(Xψ) ) holds then it turns out that both Wψ and C are Tate twists of Hodge structures of lower weight. In the K3 case, Wψ and C are Hodge structures of pure (1, 1)-type. By the Lefschetz’ theorem on (1, 1)-classes, they are generated by classes of divisors. In particular, for each of the five families the lifts to characteristic zero have Picard number at least 19, and since they form a one-dimensional family the generic Picard group has rank 19. In the second half of the paper we discuss how one can find a basis for a subgroup of finite index of the generic Picard group for the five families from [8] and for five further monomial de- formations of Delsarte quartic surfaces. For all ten families we determine H2(Yψ)G, H2(Yψ)Gmax and C as vector spaces with Frobenius action Moreover, we find curves generating C in each of the ten cases. For two families we have that Wψ is zero-dimensional. For six of the remaining eight families we manage to find curves, whose classes in cohomology generate Wψ. In the next section we prove our generalisation of the result from [8]. In the third section we discuss the quartic surface case. In Appendix A we give explicit equations for bitangents to certain particular quartic plane curves. These equations can be used to find explicit curves, generating Wψ. 2 Delsarte hypersurface Fix an integer n ≥ 2 and fix a finite field Fq. Definition 2.1. An invertible matrix A := (ai,j)0≤i,j≤n, such that all entries are nonnegative integers is called a coefficient matrix if all entries of A−1(1, . . . , 1)T are positive and each column of A contains a zero. In that case let d be an integer such that B := dA−1 has integral coefficients. We call B the map matrix. We call B(1, . . . , 1)T the weight vector, which we denote by w := (w0, . . . , wn). A vector a := (a0, . . . , an) consisting of nonnegative integers such that n∑ i=0 wiai = d holds and such that all entries of aA−1 are nonnegative is called a deformation vector. Definition 2.2. Fix a pair (A,a) consisting of coefficient matrix and a deformation vector a. Assume that gcd(q, d) = 1. Then we call (A,a) Delsarte deformation data of length n. Let (A,a) be Delsarte deformation data of length n. Let Xλ := Z  n∑ i=0 n∏ j=0 x ai,j j + λ n∏ i=0 xaii  Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 5 be the corresponding one-parameter family of hypersurfaces of weighted degree d in the weighted projective space P(w0, . . . , wn). Denote with (b0, . . . , bn) the entries of aB. Let Yλ be Z ( n∑ i=0 ydi + λ n∏ i=0 ybii ) ⊂ Pn. Remark 2.3. Our definition of w may lead to choices of the wi such that the gcd of (w0, . . . , wn) is larger than one. The choice of the wi is such that the weighted degree of the polynomial defining Xλ equals the degree of Yλ. We have a (Z/dZ)n-action on Pn induced by (g1, . . . , gn)(x0 : x1 : · · · : xn) := ( x0 : ζg1x1 : ζg2x2 : · · · : ζgnxn ) , with ζ a fixed primitive d-th root of unity. The subgroup G defined by n∑ i=1 gibi ≡ 0 mod d acts on Yλ. The rational map Pn 99K P(w) given by (y0, y1, . . . , yn) 7→ ( n∏ i=0 y b0,i i , n∏ i=0 y b1,i i , . . . , n∏ i=0 y bn,i i ) induces a rational map Yλ 99K Xλ. This rational map is Galois (i.e., the corresponding extension of function fields is Galois) and the Galois group is a subgroup of G. In particular, if all the bi,j are nonnegative then this rational map is a morphism. (This map was used by Shioda [20] to give an algorithm to calculate the Picard number of a Delsarte surface in P3.) Lemma 2.4. The hypersurface X0 is irreducible. Proof. Each column of A contains a zero by the definition of coefficient matrix. Hence xk does not divide n∑ i=0 n∏ j=0 x ai,j j for any k. Hence for every irreducible component of X0 the points such that all coordinates are nonzero are dense, and these latter points are in the image of Y0. This implies that every irreducible component of X0 is the closure of an irreducible component of the image of Y0. Since n > 1 it follows that Y0 is irreducible and hence X0 is irreducible. � Definition 2.5. We call X0 the Delsarte hypersurface associated with A and Xλ the one-dimen- sional monomial deformation associated with (A,a). If, moreover, X0 is quasismooth then we call X0 invertible hypersurface. Example 2.6. Consider x40 + x41 + x32x3 + x33x2 + λx0x1x2x3. Then A =  4 0 0 0 0 4 0 0 0 0 3 1 0 0 1 3  and a = (1, 1, 1, 1). 6 R. Kloosterman We have that B =  2 0 0 0 0 2 0 0 0 0 3 −1 0 0 −1 3  and w = (2, 2, 2, 2). In particular, we have that this family is birational a quotient of x80 + x81 + x82 + x83 + λ(x0x1x2x3) 2. The group G is generated by the automorphisms (x0, x1, x2, x3) 7→ (x0,−x1, x2, x3) and (x0, x1, x2, x3) 7→ ( x0, x1, ζ 3x2, ζx3 ) , with ζ a primitive 8-th root of unity. Definition 2.7. A hypersurface X = V (f) ⊂ Pn is in general position if V ( x0 ∂ ∂x0 f, . . . , xn ∂ ∂xn f ) is empty. Equivalently, X is smooth and for any subset {i1, . . . , ic} ⊂ {0, 1, . . . , n} we have that X ∩ V (xi1) ∩ · · · ∩ V (xic) is also smooth. Lemma 2.8. If Yλ is smooth then Yλ is in general position. Proof. Suppose we intersect Yλ with xi1 = · · · = xic = 0. If some bij is nonzero then the intersection is a Fermat hypersurface in Pn−c and is smooth. If all bij are zero then we can do the following: After a change of coordinates we may assume that {i1, . . . , ic} = {0, 1, . . . , c− 1}. We now have that Yλ is the zero set of c−1∑ i=0 xdi + h(xc, . . . , xn) for some h ∈ Fq[xc, . . . , xn]. From gcd(q, d) = 1 it follows that the singular points of the intersection V (x0, . . . , xc−1, h) are in one-to-one correspondence with the singular points of Yλ. Hence V (x0, . . . , xc−1, h) is smooth. � Recall that we started with a hypersurface Xλ ⊂ P(w) and constructed a hypersurface Yλ ⊂ Pn, such that Xλ is birational to a quotient of Yλ. Denote with Uλ := P(w) \ Xλ and Vλ := Pn \ Yλ be the respective complements. Denote now with (P(w))∗, U∗λ , V ∗λ , X∗λ, Y ∗λ , etc. the original variety minus the intersection with Z(x0 . . . xn) or Z(y0 . . . yn), the union of the coordinate hyperplanes. We have that the quotient map Pn 99K P(w) defines surjective morphisms (Pn)∗ → P(w)∗, Y ∗λ → X∗λ, V ∗λ → U∗λ . There is a second quotient map Pn → P(w) given by (z0 : · · · : zn)→ (zw0 0 : · · · : zwn n ). This map is a morphism and is a ramified Galois covering. Denote with H the corresponding Galois group. Let X̃λ be the pull back of Xλ and let Ũλ be the pull back of Uλ. Fix now a lift µ ∈ Qq of λ. Then we can define Fµ, Ũµ, Vµ, X̃µ, Yµ similarly as above. If y0, . . . , yn are projective coordinates on Pn then let Ω be( n∏ i=0 yi )( n∑ i=0 (−1)i dy0 y0 ∧ · · · ∧ d̂yi yi ∧ · · · ∧ dyn yn ) . We recall now some standard notation used to study the cohomology of a hypersurface comple- ment in Pn. Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 7 Notation 2.9. Let m = (m0, . . . ,mn) be (n+1)-tuple of positive integers, such that n∑ i=0 mi = td for some positive integer t. Then ω̃m := n∏ i=0 xmi−1 i (FA,µ(xw0 0 , . . . , xwn n ))t Ω is an n-form on the complement Ũµ of X̃µ. If we allow the mi and t to be arbitrary integers such that the equality n∑ i=0 mi = td holds then ω̃m is a form on Ũ∗µ. Let m = (m0, . . . ,mn) be (n + 1)-tuple of positive integers, such that n∑ i=0 mi = td holds for some positive integer t. Let D be the diagonal matrix dIn+1. Then ωm := n∏ i=0 ymi−1 i F tD,µ Ω is an n-form on the complement Vµ of Yµ. If we allow the mi and t to be arbitrary integers such that the equality n∑ i=0 mi = td then ωm is a form on V ∗µ . The following result seems to be known to the experts, but we include it for the reader’s convenience: Lemma 2.10. There exists a finite set S ⊂ Qq such that 0 6∈ S and for all µ ∈ Qq \ S we have that B := { ωm : 0 < mi < d for i = 0, . . . , n and n∑ i=0 mi ≡ 0 mod d } is a basis for Hn dR(Vµ,Qq). Similarly, there exists a finite set S∗ such that 0 6∈ S∗ and for all µ ∈ Qq \ S∗ we have that B∗ := { ωm : 0 ≤ mi < d for i = 0, . . . , n and n∑ i=0 mi ≡ 0 mod d } is a basis for Hn dR(V ∗µ ,Qq). Proof. The forms ωm, such that mi ≥ 1 for i = 0, . . . , n generate the de Rham cohomology group Hn dR(Vµ). By differentiating certain particular (n − 1)-forms on Vµ we have that the following relation in Hn dR(Vµ) Gyi F t Ω = tGFyi F t+1 Ω (2.1) for any form G ∈ Qq[y0, . . . , yn]td−n. (This is the so-called Griffiths–Dwork method to reduce forms in cohomology.) For µ = 0 we have that Fyi = dyd−1i . Using (2.1) we find the relation ym0 0 G(y1, . . . , yn) F t+1 0 Ω = (m0 − d+ 1)ym0−d 0 G(y1, . . . , yn) tF t0 Ω (2.2) 8 R. Kloosterman and similar relations for the other yi. In this way we can reduce forms such that all exponents are at least 0 and at most d− 1. However, if an exponent equals d− 1 then this relation yields that the class is zero in cohomology. In particular, the ωm with 0 < mi < d for i = 0, . . . , n and n∑ i=0 mi ≡ 0 mod d generate Hn dR(V0). Griffiths [11] showed that the relations of type (2.1) generate all relations and hence B is a basis for Hn dR(V0). If Xµ is smooth then the dimension of Hn dR(Vµ) is independent of µ and it is then straightforward to check that there are at most finitely many choices of µ for which Xµ is smooth and B is not a basis for Hn dR(Vµ). We now prove the statement on Hn dR(V ∗µ ). Note that if Xµ is smooth then by Lemma 2.8 it is in general position. Therefore the dimension of Hn dR(V ∗µ ) is independent of µ. Hence it suffices to show that B∗ is a basis for Hn dR(V ∗0 ). Again we have relations of type (2.1), but now we may take G ∈ Qq [ y0, y −1 0 , . . . , yn, y −1 n ] td−n. If the exponent of a variable is at most −2 then we can use the relations of the shape (2.2) to increase the exponent of this variable. However, if the exponent equals −1 we cannot do this, because then we would have to divide by zero in (2.2). In this way we obtain that B∗ generates Hn dR(V ∗0 ). Moreover, as in the above case there are no further relations and B∗ is a basis. � Remark 2.11. The µ for which B is not a basis can be determined by the methods of [18, Section 3]. Remark 2.12. Denote with Hn MW(Vλ) the n-th Monsky–Washnitzer cohomology of Vλ. The Monsky–Washnitzer cohomology is essentially the cohomology of the tensor product of de Rham complex of a lift of Vλ to characteristic zero with a weakly complete finitely generated algebra A†. The Frobenius action on the cohomology is induced by a lift of Frobenius to A†. For more details see [21, Theorem 2.4.5]. In that paper it is shown that two different lifts of Vλ yield isomorphic complexes and two choices of lifts of Frobenius yield homotopic maps on the complexes. In particular, Hn MW(Vλ) is independent of the choices made. Let µ ∈ Qq be a lift of λ. One choice of a lift of Vλ to characteristic zero is Vµ, and the construction of the Monsky–Washnitzer cohomology yields a natural map Hn dR(Vµ)→ Hn MW(Vλ) of Qq-vector spaces. If Yλ is smooth then the is an isomorphism by [1]. Since Vλ ⊂ Pn is affine and smooth we have an isomorphism Hn MW(Vλ) ∼= Hn rig(Vλ) by [2], where the latter group is rigid cohomology. Since there are infinitely many lifts µ of λ we can always choose a lift µ such that B is a basis for Hn dR(Vµ) and thereby yielding a basis for Hn MW(Vλ). If Yλ is smooth then using Lemma 2.8 we find that V ∗λ is the complement of a normal crossing divisor. In particular, we can apply [1] and find a natural isomorphism Hn MW(V ∗λ ) ∼= Hn dR(V ∗µ ). As above, we have an isomorphism Hn MW(V ∗λ ) ∼= Hn rig(V ∗λ ) and we identified a basis for Hn rig(V ∗λ ). Remark 2.13. The action of G lifts to characteristic zero. The forms ωm are eigenvectors for g∗ each element g ∈ G. Hence the G-invariant ones span Hn MW(Vλ)G. If Yλ is singular then by the definition of Monsky–Washnitzer cohomology we have that Hn MW(Vλ) is generated by expressions∑ m=(m0,...,mn),mi≥1 amωm such that there exists c1, c2 ∈ Q with c1 > 0 and v(am) ≥ c1 ( n∑ i=0 mi ) + c2. The space Hn MW(V ∗λ ) is generated by expression∑ m=(m0,...,mn),mi≥−N amωm Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 9 such that there exists c1, c2 ∈ Q with c1 > 0 and v(am) ≥ c1 ( n∑ i=0 mi ) + c2. The G-invariant subspaces are generated by similar sums, but in which only the G-invariant ωm occur. Remark 2.14. If one wants to study the Frobenius matrix by using the differential equations, like in [12] or in [8] then one needs to be more careful in lifting Vλ to characteristic zero. In [12] one has to take µ to be the Teichmüller lift of λ. The reason for this, is that a priori Frobenius maps H i ( U qµ ) to H i(Uµ). To have an operator on Hn(Uµ) we need that µq = µ. If one works directly with Frobenius on Monsky–Washnitzer chomology then this constraint on µ does not exist. From now on we use H i and H i c to indicate rigid cohomology respectively rigid cohomology with compact support. By [16, Proposition 2.1] we have canonical isomorphisms H i c(U ∗ λ) ∼= H i c(V ∗ λ )G and H i c(U ∗ λ) ∼= H i c ( Ũ∗λ )H . We want to compare the cohomology of Hn c (Vλ)G with the cohomology of Hn c (Uλ). However, Uλ may be singular, hence we work with Hn c ( Ũλ )H instead. Using Poincaré duality it suffices to compare Hn(Vλ)G with Hn ( Ũλ )H instead. Since both varieties are smooth and affine we can identify their rigid cohomology groups with their Monsky–Washnitzer cohomology groups. We will do this in order to prove: Proposition 2.15. Suppose that Hn(Vλ) → Hn(V ∗λ ) is injective. Then Hn(Vλ)G is a quotient of Hn ( Ũλ )H . In particular, the characteristic polynomial of Frobenius acting on Hn c (Vλ)G is in Q[T ] and divides the characteristic polynomial of Frobenius acting on Hn c (Uλ). Proof. Since Hn(Vλ) → Hn(V ∗λ ) is injective we have by [3] that the Poincaré dual of this map is surjective, and therefore that Hn c (Vλ)G is a quotient of Hn c (V ∗λ )G. This implies that Hn c (Vλ)G is also a quotient of Hn c ( Ũ∗λ )H . Hence it suffices to show that the kernel of natural map Hn c ( Ũ∗λ ) → Hn c ( Ũλ ) is mapped to zero in Hn c (Vλ)G. Using Poincaré duality we can consider Hn(Vλ)G as a subspace of Hn ( Ũ∗λ )H . It suffices to show that Hn(Vλ)G is in the image of Hn(Ũλ). A form ωk is in Hn(Vλ)G if and only if there is a monomial type m0 such that k = m0B. We identified Hn(Vλ)G with a subspace of Hn ( Ũ∗λ )H . The class of ωk is identified with ω̃m where m = m0(diag(w0, . . . , wn)). The entries of m are integers, which may be nonpositive. If all entries of m are positive then ωm is in the image of Hn ( Ũλ )H . Recall that B = dA−1 and therefore m0 = k1 dA. Since k has positive entries, A has positive entries and no zero column it follows that also the entries of m0 are positive and therefore all entries of m are also positive. This yields the first statement. To prove the second statement. By [16, Lemma 4.3] it follows that Hn c (Vλ)G is Frobe- nius invariant and the characteristic polynomial is in Q[T ]. Using Poincaré duality we find that Hn c (Vλ)G is a subspace of Hn c (Ũλ)H . As explained above, the latter space is isomorphic with Hn c (Uλ). � Definition 2.16. Fix Delsarte deformation data (A1,a1), . . . , (At,at) of length n. We say that they have a common cover if for every i, j we have that aT i A −1 i and aT j A −1 j are proportional. Example 2.17. Take the following five matrices 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4 ,  4 0 0 0 0 4 0 0 0 0 3 1 0 0 1 3 ,  3 1 0 0 1 3 0 0 0 0 3 1 0 0 1 3 ,  4 0 0 0 0 3 1 0 0 0 3 1 0 1 0 3 ,  3 1 0 0 0 3 1 0 0 0 3 1 1 0 0 3 . 10 R. Kloosterman In each case we take (1, 1, 1, 1)T as the deformation vector then the (Ai,ai) have a common cover. Suppose now that (A1,a1), . . . , (At,at) have a common cover. Let d be the smallest positive integer such that dA−1i has integral coefficients for all i. The sum of the entries of bi = ai ( dA−1i ) equals d. By assumption we have that for each i and j the vectors bi and bj are proportional, hence these vectors coincide and we denote this common vector with b. Denote with bj the entries of b. Denote with Xi,λ the family associated with (Ai,ai). Then Xi,λ is birational to a quotient of Yλ : n∑ i=0 ydi + λ n∏ i=0 ybii . At the beginning of this section we gave an explicit description of this map. From that description it follows that Yλ → Xi,λ is defined whenever all the yi are nonzero. We can now apply Proposition 2.15 to the above setup and we find directly that: Theorem 2.18. Let (A1,a1), . . . , (At,at) be Delsarte deformation data of length n with a com- mon cover. Denote with Xi,λ be the corresponding families of Delsarte hypersurfaces and with Yλ the common cover. Let Gi be the Galois group of the function field extension corresponding to the rational map Yλ 99K Xi,λ. Then the automorphisms in Gi extend to automorphisms of Yλ. Identify Gi with the corresponding subgroup of Aut(Yλ). Let G = G1.G2. . . . .Gt ⊂ Aut(Yλ). Suppose that Hn(Vλ) → Hn(V ∗λ ) is injective. Then for each i = 1, . . . , t we have that Hn c (Vλ)G is a quotient of Hn c (Ui,λ). In particular, the characteristic polynomial of Frobenius on Hn−1(Yλ)G is in Q[T ] and is a common factor of the characteristic polynomials of Frobenius acting on Hn−1(Xi,λ). Remark 2.19. Recall that in order to be Delsarte deformation data we need that gcd(q, (n + 1) det(Ai)) = 1 for all i. Remark 2.20. If Yλ is smooth then it is in general position by Lemma 2.8. The map Hn−1(Yλ) → Hn−1(Y ∗λ ) is injective if n − 1 is even, and has a kernel if n − 1 is odd, and this kernel is generated by the hyperplane class, see [12, Theorem 1.19]. The residue map identifies Hn(Vλ) with the primitive part of the cohomology of Hn−1(Yλ). In particular, the composition Hn(Vλ) → Hn−1(Y ∗λ ) is injective independent of the parity of n. From the diagram on [12, p. 79] it follows that the latter map factors through Hn(V ∗λ ). In particular, Hn(Vλ)→ Hn(V ∗λ ) is injective. Hence we can apply the above proposition if Yλ is smooth. The values of λ for which Yλ is singular can be determined from the formula [15, Lemma 3.7]. To conclude that there is a common factor of the zeta function is more complicated in general. The zeta function is a quotient of products of characteristic polynomials of Frobenius and there may be some cancellation in this quotient. However, if we make the extra assumptions that each Xi,λ is a hypersurface in Pn (i.e., for each i we have that w = (k, . . . , k) for some k ∈ Z>0) and we consider only values of λ for which Xλ is smooth then we have thatZ(Xi,λ, T ) n−1∏ j=0 ( 1− qjT )(−1)n = det ( I − TFrob∗ : Hn c (Ui,λ) ) . From the smoothness of Xi,λ it follows that the eigenvalues of Frobenius on Hn c (Ui,λ) have absolute value qn−1/2, hence there is no cancellation in this formula and we obtain: Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 11 Corollary 2.21. Let (A1,a1), . . . , (At,at) be Delsarte deformation data of length n with a com- mon cover. Denote with Xi,λ be the corresponding families of Delsarte hypersurfaces and with Yλ the common cover. Let Gi be the Galois group of the function field extension corresponding to Yλ → Xi,λ. Let G = G1.G2. . . . .Gt ⊂ Aut(Yλ). Suppose that for each i we have P(w) = Pn. Moreover, suppose that Yλ and each Xi,λ is smooth. Then the characteristic polynomial of Frobenius on Hn−1(Yλ)Gprim is in Q[T ] and divides the polynomialZ(Xi,λ, T ) n−1∏ j=0 ( 1− qjT )(−1)n . Remark 2.22. A complex hypersurface with quotient singularities is a Q-homology manifold and satisfies Poincaré duality. The existence of Poincaré duality is sufficient to obtain both the vanishing statement H i c(P(w) \Xi,λ) = 0 for i 6= n, 2n as well as for the purity statement on Hn c (P(w) \Xi,λ). Hence if Poincaré duality would hold for the rigid cohomology of varieties with (tame) quotient singularities over finite fields then we could extend the above corollary to the case where Xi,λ is a quasi-smooth hypersurface. We would like to compare our factor with the factor found in [8]. The groups G and G′ consists of torus automorphisms of Yλ. Let Gmax be the group of torus automorphism of Yλ. Then Gmax is an abelian group. A torus automorphism g ∈ Gmax sends Y ∗λ to itself, and descents to an automorphism of X∗λ ∼= Y ∗λ /G. Hence the quotient group Gmax/G acts on X∗λ. Since the quotient map is given by n + 1 monomials we have that a torus automorphism descends to a torus automorphism of X∗λ and U∗λ . Any torus automorphism can be extended to P(w), leaving Xλ invariant. Hence we have an action of Gmax/G on Hn(Uλ) and on Hn c (Uλ). It is straightforward to check that Gmax/G ∼= SL(FA), where SL(FA) is the group introduced in [8], and that both groups act the same. The factor from [8] is constructed as follows: The authors identify a subspace of the Dwork cohomology group Hn Dwork(Uλ)SL(FA), whose dimension equals the order of the Picard–Fuchs equation of Xλ and which is invariant under Frobenius. They show that the characteristic polynomial R′λ of Frobenius on this subspace is in K[T ] for some number field K, which can be taken Galois over Q and then take Rλ the be the least common multiple of the Galois conjugates of R′λ. To compare this polynomial with the factor constructed above, we will start by reconside- ring R′λ, i.e., we will show that it is just the characteristic polynomial of Frobenius acting on Hn c (Vλ)Gmax . Then [16, Lemma 4.3] implies that R′λ ∈ Q[T ] and that Rλ = R′λ. We start by calculating the dimension of Hn c (Vλ)Gmax . Lemma 2.23. Suppose that Xλ is Calabi–Yau, i.e., ∑ wi = d and suppose that Yλ is smooth. Then the dimension of Hn c (Vλ)Gmax equals the order of the Picard–Fuchs equation for Xλ. Proof. Since Vλ is smooth we have by Poincaré duality [3] that dimHn c (Vλ)Gmax = dimHn(Vλ)Gmax . We now calculate the latter dimension. The group Gmax consists of the (g1, . . . , gn) in (Z/dZ)n such that n∑ i=1 gibi ≡ 0 mod d. 12 R. Kloosterman From [16, Lemma 4.2] it follows that Gmax fixes the differential form ωk if and only if k ≡ tb mod d for some t ∈ Z/dZ. Hence Hn(Vλ)Gmax is spanned by ωtb where t ∈ {0, 1, . . . , d − 1} such tb mod d has no zero entry. The number of t ∈ Z/dZ such that tb mod d has a zero entry equals the number of t ∈ {0, . . . , d− 1} for which there exists an i and an integer k such that tbi = kd, or, equivalently, t d = k bi . Since 0 ≤ t < d we may assume that 0 ≤ k < bi. Using the notation from [8, Section 2] we have that the elements on the left hand side are in the set they call α and the elements on the right hand side are in the set β. In particular, the number of t such that tb has no zero entry equals d − #α ∩ β. Gährs [10, Theorem 2.8] showed that this number equals the order of the Picard–Fuchs equation. � Proposition 2.24. Suppose w0 = · · · = wn = 1, d = n + 1 and ai = 1 for i = 0, . . . , n. Then the factor R′λ found in [8] is the characteristic polynomial of Frobenius acting on Hn(Vλ)Gmax. In particular, R′λ ∈ Q[T ] and Rλ = R′λ. Proof. Since P(w) = Pn we have Uλ = Ũλ. Hence we can discuss differential forms on the complement of Xλ. The factor R′λ(T ) obtained in [8] using the p-adic Picard–Fuchs equation in Dwork cohomo- logy. The main result from [12] yields a differential equation satisfied by the Frobenius operator on Hn MW(Uλ,Qq) and that this differential equation can also be found using Dwork cohomology. In particular, R′λ(T ) is the characteristic polynomial of Frobenius acting on the subspace P containing ωa and invariant under the Picard–Fuchs operator. This subspace P is contained in the span of {ωsa : s = 1, 2, . . . }. Pick a form ω̃ta restrict this form to U∗λ and then pull it back to a form on V ∗λ . Then this pull back is ωtb. This form is defined on all of Vλ. Hence the pullback of P to Hn(Vλ) is well-defined and is contained in Hn(Vλ)Gmax . Hence P is a subspace of Hn(Vλ)Gmax . Since both spaces have the same dimension by Lemma 2.23 they coincide, i.e., P ∼= Hn(Vλ)Gmax as vector spaces with Frobenius action. Now Rλ is the characteristic polynomial of qn Frob−1 acting on P . Using Poincaré duality this equals the characteristic polynomial of Frobenius acting on Hn c (Vλ)Gmax . This yields the first claim. The obtained polynomial is in Q[T ] by [16, Lemma 4.3] and hence R′λ(T ) = Rλ(T ). � 3 Case of quartic surfaces In this section we consider the case of invertible quartic polynomials. Up to permutation of the coordinates there are 10 invertible quartic polynomials in four variables. For each of these quartics we take a = (1, 1, 1, 1) as the deformation vector. In Fig. 1 we list the 10 families, which we denote here with X (i) λ . We provide the following information in the table. In the column “d” we list the minimal degree of a Fermat cover of the central fiber. When we discuss one of the examples we always assume that gcd(q, d) = 1. In the next column we list the deformation vector b := (1, 1, 1, 1)TB. Hence the corresponding Fermat cover Y (i) λ is defined by xd0 + xd1 + xd2 + xd3 + λxb00 x b1 1 x b2 2 x b3 3 . Let G be the Galois group of the function field extension corresponding to the morphism Y ∗λ → X∗λ. The next two columns deal with H3(Vλ)G, for λ such that Yλ is smooth. In the Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 13 i F0 d (1, 1, 1, 1)TB PF dimWλ c 1 x40 + x41 + x42 + x43 4 (1, 1, 1, 1) 3 18 0 2 x40 + x41 + x2x3 ( x22 + x23 ) 8 (2, 2, 2, 2) 3 12 6 3 x0x1 ( x20 + x21 ) + x2x3 ( x22 + x23 ) 8 (2, 2, 2, 2) 3 10 8 4 x40 + x1x 3 2 + x2x 3 3 + x3x 3 1, 28 (7, 7, 7, 7) 3 18 0 5 x0x 3 1 + x1x 3 2 + x2x 3 3 + x3x 3 0 80 (20, 20, 20, 20) 3 16 2 6 x40 + x41 + x32x3 + x43 12 (3, 3, 4, 2) 6 12 3 7 x0x1 ( x20 + x21 ) + x32x3 + x43 24 (6, 6, 8, 4) 6 8 7 8 x30x1 + x41 + x32x3 + x43 12 (4, 2, 4, 2) 4 12 5 9 x40 + x31x2 + x32x3 + x43 36 (9, 12, 8, 7) 18 0 3 10 x30x1 + x31x2 + x32x3 + x43 108 (36, 24, 28, 20) 18 0 3 Figure 1. The 10 families X (i) λ . column PF we list the dimension of H3(Vλ)Gmax . We calculated this entry as follows: from the results from [16, Section 4] it follows that a basis for this vector space consists of those ωk such that all entries of k are between 1 and d− 1 and there is a t ∈ Z such that k ≡ tb mod d. It is straight forward to determine the number of these k. As discussed in the previous section, this number equals the order of the Picard–Fuchs equation of Xi λ. The next column concerns the subspace W (i) λ ⊂ H3 ( V (i) λ )G . The ωk such that each of the entries of k is in {1, . . . , d− 1} and there exists a vector m ∈ Z4 such that k ≡mA mod d form a basis for H3 ( V (i) λ )G . For each of the 10 examples we checked for each k if such a m existed or not and used this to calculate dimW (i) λ = dimH3 ( V (i) λ )G − dimH3 ( V (i) λ )Gmax . For the families 1, 2, 3, 6, 7 we listed all these k in Fig. 2 (they are enlisted in the corresponding column in Fig. 2, in the first column there is a choice for a possible m, the forms marked with (PF ) are in H3(Vλ)Gmax). For i = 4, 5, 8, 9, 10 we will describe W (i) λ in the examples below. Finally, the column c then equals 21 − dimH3(Vλ)G. As we argued in the introduction the subspace C of dimension c and W (i) λ are generated by classes of curves on X (i) λ . For the families i = 1, 2, 3, 6, 7, 9, 10 we give a recipe to find linear combinations of curves on X (i) λ , which generate C and W (i) λ . In fact, for all i we have that C is generated by curves, each of which is contained in one of the coordinate hyperplanes. These curves are easy to find for each i. For i = 9, 10 we have that W (i) λ = 0. For i = 1, 2, 3, 6, 7 we can find various del Pezzo surfaces of degree 2 together with morphisms of degree 2, such that linear combinations of pull backs of curves from these del Pezzo surfaces generated W (i) λ . For i = 5 we have a similar procedure using del Pezzo surfaces of degree 5. The first five families have a single common cover, also the sixth and seventh family have a common cover. The common factor of the first five examples has degree 3. However, the first three examples have a common factor of degree 5 and the first and the second example have a common factor of degree 7. The following proposition now shows that claim about W (i) λ for i = 1, 2, 3, 6, 7: Proposition 3.1. Consider one of the families X (i) λ with i ∈ {1, 2, 3, 6, 7} from Fig. 1. Then there exist families of del Pezzo surfaces S (i,j) λ and degree 2 morphisms ϕ (i,j) λ : X (i) λ → S (i,j) λ such that if i ∈ {1, 6, 7} then for almost all λ we have that W (i) λ ⊂ ∑ j ϕ (i,j)∗ λ ( H2 ( S (i,j) λ )) 14 R. Kloosterman m 3.1 3.2 3.3 3.6 3.7 A = 10, B = 11 1111 1111(PF ) 2222(PF ) 2222(PF ) 3342(PF ) 6, 6, 8, 4(PF ) 1124 − − − 338A(PF ) 6, 6, 16, 20(PF ) 1133 1133 2266 2266 − − 1214 − − − 364B 3, 15, 8, 22 1223 1223 2437 1537 3687 3, 15, 16, 14 1232 1232 2473 1573 − − 1313 1313 − − 3948 − 1322 1322 2644 − 3984 − 1331 1331 − − − − 2114 − − − 634B 15, 3, 8, 22 2123 2123 4237 5137 6387 15, 3, 16, 14 2132 2132 4273 5173 − − 2222 2222(PF ) 4444(PF ) 4444(PF ) 6684(PF ) 12, 12, 16, 8(PF ) 2213 2213 − − 6648(PF ) 12, 12, 8, 16(PF ) 2231 2231 − − − − 2312 2312 4615 3715 6945 9, 21, 8, 10 2321 2321 4651 3751 6981 9, 21, 16, 2 3113 3113 − − 9348 − 3122 3122 6244 − 9384 − 3131 3131 − − − − 3212 3212 6415 7315 9645 21, 9, 8, 10 3221 3221 6451 7351 9681 21, 9, 16, 2 3311 3311 6622 6622 9942(PF ) 18, 18, 8, 4(PF ) 3324 − − − 998A(PF ) 18, 18, 16, 20(PF ) 3333 3333(PF ) 6666(PF ) 6666(PF ) − − Figure 2. Generators for H3 ( V (i) λ )G . and if i ∈ {2, 3} then for almost all λ we have that W (i) λ ∩ ∑ j ϕ (i,j)∗ λ ( H2 ( S (i,j) λ )) has codimension 2 in W (i) λ and the forms ω̃1133, ω̃3311 generate a complementary subspace in W (i) λ . If i = 3 or i = 7 or q ≡ 1 mod 4 then we can take the S (i,j) λ to be defined over Fq. If q ≡ 3 mod 4 and i ∈ {1, 2, 6} then some of the S (i,j) λ are only defined over Fq2. Proof. Note that W (i) λ is spanned by ω̃m where m are precisely these entries from the first column of Fig. 2 such that in the column corresponding to i there the entry is different from “−” and is without the mark “(PF )”. Note also that in the notation of the previous section we have Ũ (i) λ = U (i) λ . Hence we denote differential forms on the complement of X (i) λ with ω̃ and forms on the complement of Y (i) λ with ω. A defining polynomial for X (i) 0 can be found in Fig. 1. Recall that for each family we took (1, 1, 1, 1) as the deformation factor. In particular, each of the five families under consideration is each invariant under the automorphisms σ and τ defined by σ(x0, x1, x2, x3) := (x1, x0, x2, x3), τ(x0, x1, x2, x3) := (−x1,−x0, x2, x3). Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 15 A straightforward calculation shows that the quotients of X (i) λ by σ and by τ are both surfaces of degree 4 in P(1, 1, 1, 2), and that for general λ they are smooth (explicit equations for these surfaces can be found in the appendix). Hence the quotient surfaces are del Pezzo surfaces of degree 2. Denote the corresponding surfaces with S (i,1) λ and S (i,2) λ Let m = (a, b, c, d) with a, b, c, d ∈ {1, 2, 3, 4} be such that a + b + c + d ≡ 0 mod 4. Then ω̃m+σ∗ω̃m is invariant under σ∗ and therefore contained in π∗1 ( H2 ( S (i,1) λ )) . Similarly, ω̃a+τ∗ω̃a is contained in π∗2 ( H2 ( S (i,2) λ )) . Now ω̃m + σ∗ω̃m = ω̃abcd − ω̃bacd and ω̃m + τ∗ω̃m = ωabcd + (−1)a+b+1ωbacd. Hence, if a+ b is odd then ω̃m ∈ π∗1 ( H2 ( S (i,1) λ )) + π∗2 ( H2 ( S (i,2) λ )) . In the case i = 7 we have that W (i) λ is generated by forms ω̃m with a+ b odd and we finished this case. In the case i = 3 we have that W (i) λ is generated by forms with a+ b odd and the two forms ω̃1133 and ω̃3311. Hence we finishes also this case. In the remaining cases i = 1, 2, 6 we have a further automorphism τ1 : (x0, x1, x2, x3) 7→ (Ix1,−Ix0, x2, x3), where I2 = −1. Denote S3,i λ the quotient by τ1. If q ≡ 1 mod 4 then S3,i λ is defined over Fq, but if q ≡ 3 mod 4 then it is only defined over Fq2 . Note that τ∗1 (ω̃abcd) = (−1)b+1(I)a+bω̃bacd. Hence if b is odd and a + b ≡ 0 mod 4 then ω̃abcd + ω̃bacd is fixed under τ∗1 and, as above, we find that ω̃abcd is in π∗1(H2(Si,1λ )) + π∗3(H2(Si,3λ )). Using Fig. 2 we can conclude that we recovered any ω̃m such that the first two entries are distinct. This finishes the proof for the case i = 6. In the case i = 2, we only miss the forms ω̃1133 and ω̃3311, hence we are also done in this case. In the case i = 1 there is a S4 symmetry we can use. We recover all ω̃k with at least two distinct entries in k and this finishes also this case. � Remark 3.2. In the cases i = 2, 3 we do not recover ω̃1133 and ω̃3311. However, the families i = 1, 2, 3 have x240 + x241 + x242 + x243 + λ (x0x1x2x3) 6 as a common cover. For each of three families the form ω̃1133 is pulled back to the form ω6,6,18,18 on Vλ. Hence we can use X (1) λ to express this form in terms of divisors pulled back form S1,3 λ . In the following examples we discuss how to find generators for the subspace C. For the examples i = 4, 5, 8 we list a basis for H3 ( V (i) λ )Gmax and also discuss strategies to find generators for Wλ. Example 3.3. For the case i = 1, 2, 3, 6, 7 we note that Proposition 3.1 yields a basis for W (i) λ in terms of curves pulled back from del Pezzo surfaces S(i,j). In the appendix we will explain how to find these curves. For each of these cases we can find generators for C in each of these cases, but the approach depends on i: i = 1 C = 0 in this case. i = 2 The curves given by x3 = 0, x0 = −Ikx1 and the ones given by x4 = 0, x0 = −Ikx1 are in C, with I2 = −1. One easily checks that they generate C. These curves can also be obtained by pulling back curves from the del Pezzo quotients: For example, consider the quotient by the automorphism σ : (x0, x1, x2, x3) 7→ (x1, x0, x2, x3). We find that 16 R. Kloosterman ωabcd−ωbacd is a 1 eigenvector if (a, b) 6= (b, a). However, there are only 5 such eigenvectors. The Picard group of the del Pezzo surface has rank 8. The additional three divisors are the hyperplane class and the two curves pulled back from the curves x3 = x21 +x22 = 0 and x4 = x21 + x22 = 0. i = 3 We have that for j < 2 and k > 1 the line xj = xk = 0 is contained in X (3) λ as are xm = 0, x2 = ±Ix3, m ≤ 2 and xm = 0, x1 = ±Ix0, m ∈ {2, 3}. These are 12 curves, but generate a rank 9 sublattice of the Picard lattice, and this lattice contains the hyperplane class. Linear combinations of these curves span C. i = 6 As in the case i = 2 in this case we have that the automorphism σ fixes only six eigenvectors of the form ω̃k − ω̃σ(k). The seventh eigenvector is the class of the curve x3 = 0, x20 + x21, which is an element of C. The other coordinate hyperplanes yields three further curves, contributing another two to the Picard number. i = 7 In this case we take x0 = 0 or x1 = 0 then we find x3(x 3 2 + x33) = 0. In this way we find 8 lines, contributing six to cohomology. We discuss now the other five examples: Example 3.4. Consider now the case i = 4. In this case the Fermat cover Y (4) λ has degree 28. Let G be the associated group of torus automorphisms. Then the multiples ωk with k a multiple of (7, 7, 7, 7) generate a rank 3 subspace of H3 ( V (4) λ )G which is common to the examples i = 1, 2, 3. The other monomial types associated with forms in H3(V (4) λ )G are (7, 11, 15, 23), (7, 3, 27, 19), (14, 2, 18, 22), (14, 6, 26, 10), (21, 1, 9, 25), (21, 5, 17, 13) and those obtained by a cyclic permutation of the last three coordinates. In particular, these ωk generate already a rank 21 subspace of H3(Vλ)G, which has dimension at most 21. Hence we found a basis for H3 ( V (4) λ )G and we have C = 0 in this case. One can obtain some information on the zeta function as follows. If q ≡ 1 mod 28 then we can factor the zeta function over Qq according to strong equivalence classes (cf. [16, Section 4]). The strong equivalence class of (7, 7, 7, 7) consists further of (14, 14, 14, 14) and (21, 21, 21, 12). The class of (7, 11, 23, 15) consists further of (14, 18, 2, 22) and of (21, 25, 9, 1). The class of (7, 3, 19, 27) consists further of (14, 10, 26, 6) and of (21, 17, 5, 13). The other classes can be obtained by permutation the last three coordinates. In particular, we find that the characteristic polynomial on H3 ( V (4) λ )G can be written as P3Q 3 3R 3 3, where P3, Q3 and R3 are in Qq[T ] and have degree 3. The polynomial P3 is the common factor and by Corollary 2.21 in Q[T ]. Since P3Q 3 3R 3 3 ∈ Q[T ] we find that also Q3(T )R3(T ) is in Q[T ]. The results from [15, Section 5] yield three explicit matrices, each 3×3, whose entries are ratio- nal functions of generalised p-adic hypergeometric functions, such that the three corresponding characteristic polynomials are P3, Q3 and R3. As mentioned in the introduction of this paper, we did not find a complete set of generators for generic Picard group for two of the ten families. This family is one of these two families. Example 3.5. The degree of the Fermat cover of the fifth example is 80. The monomial type (20, 20, 20, 20) and its two multiplies in H3 ( V (5) λ ) yield the factor common with the examples i = 1, 2, 3, 4. The other monomial types associated with classes in H3 ( V (5) λ )G , are (4, 52, 36, 58), (24, 72, 56, 8), (44, 12, 76, 28), (64, 32, 16, 48) and the cyclic permutations of these. Hence W (5) λ has dimension 16. The subspace C has dimension 2 and contains the classes of the lines x0 = x2 = 0 and x1 = x3 = 0. Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 17 To find the curves contributing to the rank 16 part, we can use permutations, similarly as in the above examples. The cyclic permutation of 1, 2, 3, 4 is odd. Denote this permutation by σ0. The quotient by this permutation is a del Pezzo surface of degree 5. Fix now a primitive fifth root of unity ζ. Let ρ := (x0, x1, x2, x3) 7→ ( ζx0, ζ 3x1, ζ 4x2, ζ 2x1 ) . For i = 1, 2, 3, 4 we set σi = σ0ρ i. Then each σi has order 4. Let m be a monomial types such that ω̃m is pulled back to one of the 16 forms H3 ( V (5) λ )G not a multiple of (20, 20, 20, 20). Consider now { 3∑ j=0 σji ω̃m : i = 0, . . . , 3 } . A direct calculation using a Vandermonde deter- minant shows that these four forms are linearly independent and that their span contains ω̃k. Hence ω̃k is contained in the subspace spanned by 4 ∪ i=1 H3(Uλ)σi . So each of the ω̃k can be expressed as a linear combination of curves on the del Pezzo surface X (5) λ /σi, with i = 0, 1, 2, 3. Using the terminology of [15, Section 6] we have two weak equivalence classes of monomial types, one consisting of three monomial types and consisting of 16 monomial types. The large class decomposes in four strong equivalence classes. These four strong equivalence classes are in one σ-orbit. From this we obtain if q ≡ 1 mod 20 then the characteristic polynomial of Frobenius is P4Q 4 4, where both P4 and Q4 are of degree 4. A different approach to find curves on X (5) λ would be to use the line x0 = x2 = 0 to find an elliptic fibration. A Weierstrass equation for this fibration is y2 = x3 − 27s4 ( λ4 + 144 ) x− 54s ( −s5λ6 + 864s10 + 648s5λ2 + 864 ) . For general λ the fibration has 2 fibers of type II and 20 fibers of type I1. The sections of this fibration and the fiber class generate the Picard group for general λ. Example 3.6. For i = 8 that the Fermat cover has degree 12, and the deformation vector is pulled back to 4242. The Picard–Fuchs equation has order 4. The other monomial types ωk are built up from pairs from 42, 45, 48, 4B, 81, 84, 87, 8A (where A = 10, B = 11) such that the entries add up to a multiple of 12 and such that k is not a multiple of 4242. In total we find 12 such forms. The complementary five-dimensional subspace comes from coordinate plane sections, i.e., x1 = 0 yields x3 ( x32 + x33 ) , and also x3 = 0 contributes. The total contribution is 5. We do not have any odd permutation to work with. However, this surface has many elliptic fibrations and one may be able to work with them. As mentioned in the introduction of this paper, we did not find a complete set of generators for generic Picard group for two of the ten families. This family is one of these two families. Example 3.7. In the ninth example we have that the common cover has degree 36. The deformation monomial has exponents 9, 12, 8, 7. There are 18 multiples of this vector without a zero in Z/36Z. Hence the Picard–Fuchs equation has degree 18. Moreover, the curves x2 = 0, x0 = ikx4 together with the hyperplane class generate the generic Picard group. Example 3.8. In the tenth examples we have that the Fermat cover has degree 108. The deformation monomial has exponents 36, 24, 28, 20. There are 18 multiples of this vector without a zero in Z/108Z. Hence the Picard–Fuchs equation has degree 18. Moreover, we have the curves x1 = x3 = 0, x1 = 0, x2 = ωix3 and x3 = 0, x30 − x21x2 = 0. This are five curves admitting two relations. 18 R. Kloosterman Remark 3.9. In two cases we did not find generators. In these two cases different there is no permutation σ of the coordinates which is automorphism of the family and such that the quotient surface is a rational surface. In the other examples with nontrivially Wλ, this space was generated by pull backs of curves coming from rational surfaces. It is the author’s experience that in characteristic zero, establishing explicit curves generating the Picard group of a surface, is an easier problem when working with surfaces with h2,0 = 0 then when working with surfaces with h2,0 > 0. This can be partly explained by the fact that degrees and intersection numbers of generators of the Picard group are determined by the topology of the surface in the case h2,0 = 0, but not in the case h2,0 > 0. A similar problem is determining a basis of the Mordell–Weil group of an elliptic K3 surfaces (which is equivalent to determining generators for the Nèron–Severi group of that surface). This turned out to be much simplified if the K3 surface is in various ways the pull back of a rational elliptic surface. (E.g., see [7, 14, 19].) A Bitangents to special plane quartics In Section 3 we considered ten pencils of quartic surfaces. In Proposition 3.1 we showed that five of these pencils are (each in multiple ways) double covers of pencils of del Pezzo surfaces of degree two and we showed how the knowledge of the Picard group of these del Pezzo surfaces is sufficient to determine the generic Picard group of each pencil. In this section we explain how one can find explicit generators for the Picard group of these del Pezzo surfaces. It is well-known that such a surface is a double cover of P2 ramified along a quartic curve. If the quartic curve is smooth then its has 28 bitangents. These bitangents are pulled back to two lines on the del Pezzo surface, and these lines generate the Picard group. In order to find explicit equations for the del Pezzo surfaces of degree 2 and the quartic curves we are going to make the steps from the proof of Proposition 3.1 explicit. This proposition applies only to X (i) λ with i ∈ {1, 2, 3, 6, 7}, hence we concentrate on these cases. To ease the calculations we start by decomposing the defining polynomials for X (i) λ in sums of two polynomials. Therefore define the following polynomials f1(x0, x1, x2, x3) := x40 + x41 + λx0x1x2x3, f2(x0, x1, x2, x3) := x0x1 ( x20 + x21 ) + λx0x1x2x3, g1(x2, x3) := x42 + x43, g2(x2, x3) := x2x3 ( x22 + x23 ) , g3(x2, x3) := x32x3 + x43, h1(u, v, x2, x3) := u4 − 4u2v + 2v2 + λvx2x3, h2(u, v, x2, x3) := v ( u2 − 2v ) + λvx2x3. The five pencils of quartic surfaces under consideration are defined by the vanishing of f1 + g1, f1 + g2, f2 + g2, f1 + g3, f2 + g3. A.1 S (i,1) λ As we noted in the proof of Proposition 3.1 each of these families is invariant under the auto- morphism σ : (x0, x1, x2, x3) 7→ (x1, x0, x2, x3). In particular, each of the defining polynomials is also a polynomial in x0 + x1, x0x1, x2, x3. We defined h1, h2 such that hj(x0 + x1, x0x1, x2, x3) = fj(x0, x1, x2, x3). Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 19 Therefore the quotient S (i,1) λ of X (i) λ by σ is the zeroset of h1 + g1, h1 + g2, h2 + g2, h1 + g3, h2 + g3 in P(1, 2, 1, 1). These polynomials define five families of surfaces in P(1, 2, 1, 1). The general member is a del Pezzo surface of degree 2. The rational map P(1, 2, 1, 1) 99K P2 defined by (u : v : x2 : x3)→ (u : x2 : x3) is defined on all of S (i,1) λ . It establishes this surfaces as a double cover of P2 ramified along the zeroset of qi, the discriminant qi of the defining polynomial of S (i,1) λ considered as polynomial in v. These discriminant are straightforward to compute. We list them here: q1 := −8x42 + λ2x22x 2 3 − 8λx2x3u 2 − 8x43 + 8u4, q2 := −8x32x3 + λ2x22x 2 3 − 8x2x 3 3 − 8λx2x3u 2 + 8u4, q3 := 8x32x3 + λ2x22x 2 3t+ 8x2x 3 3 + 2λx2x3u 2 + u4, q6 := −8x42 + λ2x22x 2 3 − 8x32x3 − 8λx2x3u 2 + 8u4, q7 := 8x42 + λ2x22x 2 3 + 8x32x3 + 2λx2x3u 2 + u4. Our aim is to find the bitangents to these curves and then pull them back to X (i) λ . If λ is chosen such that the quartic curve is smooth then there are 28 bitangents. We start by looking for bitangents of the shape u = a2x2 + a3x3. Such a line is a bitangent to the curve qi = 0 if we can find further b, c such that the following polynomial vanishes q1(a2x2 + a3x3, x3, x2)− 8 ( a42 − 1 )( x22 + bx2x3 + cx23 )2 if i = 1, q2(a2x2 + a3x3, x3, x2)− 8a42 ( x22 + bx2x3 + cx23 )2 if i = 2, q3(a2x2 + a3x3, x3, x2)− a42 ( x22 + bx2x3 + cx23 )2 if i = 3, q6(a2x2 + a3x3, x3, x2)− 8a42 ( x22 + bx2x3 + cx23 )2 if i = 6, q7(a2x2 + a3x3, x3, x2)− a42 ( x22 + bx2x3 + cx23 )2 if i = 7. The factors 8 ( a42 − 1 ) , 8a42 and a42 are chosen in order to kill the coefficient of x43 in each of the polynomials. Hence each of the five above polynomials is a polynomial of degree 3 in x3. These polynomials can be computed with the help of some computeralgebra package. Unfortunately, the obtained expressions are too long to include them here. From these calculations one deduces that both the coefficient of x33 and of x23 are linear in b and c. We can solve for b and c and substitute the result. However, in order to solve for b and c we have to divide by a42 − 1 if i = 1 and by a2 in the other cases, hence for the moment we have to assume that they are nonzero. We are then left with two nonzero coefficients. The coefficient of x03 is a cubic in a3. Elimi- nating a3 leaves a polynomial of degree either 20 (if i = 1) or 24 (if i 6= 1) in a2, which we list below. Each of the zeroes yields a possible value for a2. One easily checks that each value for a2 determines a unique value for a3. In this way we find 20 or 24 bitangents. Note that each of the five families admits the automorphism (u, v, x2, x3) 7→ (u, v,−x2,−x3). This implies that if (a2, a3) defines a bitangent then so does (−a2,−a3). Hence the final polynomial in a2 is actually a polynomial in a22. Depending on the case there are further automorphisms, which could give further simplifications. We now list for each case the degree 24 polynomial in a2. The case i = 1 is slightly more involved then the other ones, so we start with the case i ≥ 2. For i = 2 we find that if a2 is a zero of(( λ2 + 16 ) a42 + 4λa22λ+ 2 )(( λ2 − 16 ) a42 + 4λa22λ+ 2 ) · · · 20 R. Kloosterman · · · ( 1024a162 + 128λ3a142 + ( 2λ6 + 960λ2 ) a122 + ( 20λ5 + 2560 ) a102 + · · · · · ·+ ( 73λ4 + 2176 ) a82 + 120λ3a62 + 92λ2a42 + 32λa22 + 4 ) then there is a unique a3 yielding a bitangent. The degree 16 factor can be written as the product of two factors of degree 8 over Fq (√ 2 ) . This yields 24 of the 28 bitangents. For i = 3 we find that a2 is a zero of( 2a42 + ta22 + 2 )( 2a42 − ta22 − 2 ) · · · ( a82 + 4a62 + ( λ2 − 4λ+ 8 ) a42 + (4λ− 8)a22 + 4 ) · · · · · · ( a82 − 4a62 + ( λ2 + 4λ+ 8 ) a42 + (4λ+ 8)a22 + 4 ) then there is a unique a3 yielding a bitangent. Each of the two degree 8 factors is a product of two factors of degree 4 factors over Fq (√ −1 ) . This yields 24 of the 28 bitangents. For i = 4 we find that a2 is a zero of( −512λ6 − 1769472 ) a242 − 9216λ5a222 + ( 2λ10 − 62976λ4 ) a202 + · · · · · · ( 36λ9 − 286720λ3 ) a182 + ( 273λ8 − 663552λ2 ) a162 + · · · · · ·+ ( 1136λ7 − 700416λ ) a142 + ( 2840λ6 − 276480 ) a122 + 4416λ5a102 + · · · · · ·+ 4312λ4a82 + 2624λ3a62 + 960λ2a42 + 192λa22 + 16 then there is a unique a3 yielding a bitangent. This yields 24 of the 28 bitangents. For i = 5 we find that a2 is a zero of( −3a82 + 2λa62 + λ2a42 + 12a42 + 4λa22 + 4 ) · · · · · · ( 9a162 + 6λa142 + ( 7λ2a122 + 36 ) a122 + ( 60− 2λ3 ) a102 λ+ · · · · · ·+ ( λ4 − 20λ2 + 156 ) a82 + ( 8λ3 − 56λ ) a62 + ( 24λ2 − 48 ) a42 + 32a22λ+ 16 ) then there is a unique a3 yielding a bitangent. This yields 24 of the 28 bitangents. If ω2 = −ω−1 then over Fq(ω) we can write the degree 16 factor as a product of two factors of degree 8. To finish the cases i = 2, 3, 6, 7 we need to find 4 further bitangents. The above approach gives all bitangents of the form u = a2x2 + a3x3 with a2 6= 0. It turns out that there are no bitangents with a2 = 0, however there are bitangents of the form a2x2 + a3x2 = 0. One easily sees that the line x3 = 0 is a hyperflex line (and therefore a bitangent) and that the remaining three bitangents are of the form x2 = ax3, with a a zero of a ( 8a2 + λ2a+ 8 ) if i = 2, a ( a2 + 1 ) if i = 3, 8a3 + λ2a2 + 2 if i = 6, (a+ 1) ( a2 − a+ 1 ) if i = 7. In the cases i = 2, 3, 6, 7 we find that 24 of the bitangents can be described in terms of a polynomial in a2 of degree 24. Since a1 is the unique root of a polynomial with coefficients in Fq(λ, a2) we find that a1 ∈ Fq(λ, a2). The equations defining c and d are linear, hence they are also in Fq(λ, a2). Hence the bitangent is defined over Fq(λ, a2). However, the lines on the del Pezzo surface may be defined over a degree 2 extension. If ` = V (−u + ax2 + bx3) then the equation for the del Pezzo surface is a quadratic equation in v. It restriction to u = a1x2 + a2x3 is a quadratic equation with discriminant qi\|`. This discriminant is of the form Ci ( x22 + bix2x3 + cix 2 3 )2 . Hence to define each of the two corresponding lines on S (1,i) λ we need to take a square root of C. An explicit calculation now show that C depends only on i and a2. More precisely, we have that Ci equals 8a42, a 4 2, 8a42, a 4 2 for i = 2, 3, 6, 7. Hence for i = 3, 7 both lines are defined over Fq(λ, a2) but for i = 2, 6 they are defined over Fq ( λ, a2, √ 2 ) . Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces 21 Similarly, one easily checks that the flex line is defined over Fq and that the two corresponding lines on the del Pezzo surface are defined over Fq if i = 3, 7 and over Fq (√ −1 ) if i = 2, 6 and that each of the remaining lines are defined over Fq(a) if i = 3, 7 and Fq (√ 2, λ, a ) if i = 2, 6. For i = 1 we can copy the above approach, but in the first step we find 20 rather than 24 bitangents of the form u = a2x2 + a3x3. These 20 bitangents are one of the following (where I is a fixed root of −1): 1) a3 = a2 and a42 ( λ2 + 16 ) − 16λa22 + λ2 + 16 = 0, 2) a3 = −a2 and a42 ( λ2 + 16 ) + 16λa22 + λ2 + 16 = 0, 3) a3 = Ia2 and a42 ( λ2 − 16 ) − 16Iλa22 + λ2 − 16 = 0, 4) a3 = −Ia2 and a42 ( λ2 − 16 ) − 16Iλa22 + λ2 − 16 = 0, 5) a3 = t 4a2 and a43 = 1. Using symmetry we find that further bitangents are given by a3 = t 4a2 and a42 = 1. There are four further bitangents of the form x2 = ax3 with 8a4 + λ2a2 + 8 = 0. One easily checks that the corresponding lines on the del Pezzo surface are defined over the field Fq ( λ, a2, √ 2(a42 − 1) ) (if a42 6= 1), over Fq ( λ, a1, √ 2(a43 − 1) ) (if a43 6= 1) and over Fq ( λ, a, √ 2 ) (for the final four lines). A.2 S (i,2) λ and S (i,3) λ Once we found the lines on S (i,1) λ we can use them to find also the lines on S (i,2) λ and S (i,3) λ . Let h3(u, v, x2, x3) := u4 + 4u2v + 2v2 + λvx2x3, h4(u, v, x2, x3) := v ( u2 + 2v ) + λvx2x3. Proceeding as above we find that that S (i,2) λ is defined by h3 + g1, h4 + g2, h4 + g2, h3 + g3, h4 + g3. The map (u, v, x2, x3) 7→ (uI, v, x2, x3) defines an isomorphism S (i,2) λ → S (i,1) λ for i = 1, 2, 6. The map (u, v, x2, x3) 7→ (uI,−v, Ix2, Ix3) defines an isomorphism S (i,2) λ → S (i,1) λ for i = 3, 7. Let h5(u, v, x2, x3) := u4 − 4Iu2v − 2v2 + λvx2x3 then S (i,3) λ is defined (for i = 1, 2, 6) by h5 + g1, h5 + g2, h5 + g3. The map (u, v, x2, x3) 7→ (u, Iv, x2, Ix3) defines an isomorphism S (i,3) λ → S (i,2) λ for i = 1. The map (u, v, x2, x3) 7→ ( u, Iv, ζx2, ζ 5x3 ) defines an isomorphism S (i,3) λ → S (i,2) λ for i = 2. For i = 7 we need also to act on λ: The map (u, v, x2, x3) 7→ (u, Iv, x2, x3) defines an isomorphism S (i,3) −Iλ → S (i,2) λ for i = 2. Substituting v = x0x1 and u = x0 +x1 (if j = 1), u = x− y (if j = 2) or u = x+ Iy (if j = 3) in the equations of a line on S (i,j) λ then yields the corresponding conic on X (i,j) λ . Acknowledgements The author would like to thank John Voight and Tyler Kelly for various conversations on this topic. The author would like to thank the referees for various suggestions to improve the exposition. 22 R. 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(1986), 33–59. https://doi.org/10.1016/S0764-4442(97)88895-7 https://doi.org/10.1016/S0764-4442(97)88895-7 https://doi.org/10.1515/ADVGEOM.2011.029 https://arxiv.org/abs/0905.2099 https://doi.org/10.1090/S1056-3911-2011-00544-4 https://doi.org/10.1090/S1056-3911-2011-00544-4 https://arxiv.org/abs/0809.1791 https://arxiv.org/abs/hep-th/0402133 https://arxiv.org/abs/math.NT/9911274 https://arxiv.org/abs/1612.09249 https://doi.org/10.1007/BF02684886 https://arxiv.org/abs/1109.3462 https://doi.org/10.2307/1970747 https://doi.org/10.1007/BF02698924 https://doi.org/10.4310/ATMP.2013.v17.n6.a8 https://arxiv.org/abs/1304.3417 https://arxiv.org/abs/math.AG/0502017 https://doi.org/10.2140/ant.2007.1.421 https://doi.org/10.2140/ant.2007.1.421 https://arxiv.org/abs/math.NT/0703120 https://doi.org/10.1112/S0024611503014461 https://doi.org/10.1007/s10208-014-9242-8 https://arxiv.org/abs/1307.1250 https://doi.org/10.2307/2374678 https://doi.org/10.2307/2374678 https://doi.org/10.24033/msmf.324 1 Introduction 2 Delsarte hypersurface 3 Case of quartic surfaces A Bitangents to special plane quartics A.1 S(i,1) A.2 S(i,2) and S(i,3) References

Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Institution

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