The Schottky problem in genus 5 Charles Siegel March 31, 2012 1 Introduction 4 Incidence Relations 5 Degenerations of abelian varieties The Schottky problem is a classical problem in algebraic geometry. Let C be a We use configurations of points and lines over finite fields to study the structure To see that deg β = 119, degenerate to the boundary, where results from [Iza91] 1 curve of genus g, and α1; : : : ; αg; β1; : : : ; βg a symplectic basis for H (C; Z). Then of the fibers of maps. Given a configuration V of this type, we can study con- and [Don87] imply that there is a component where the map is generically un- R we can choose a basis for holomorphic 1-forms !1;:::;!g such that !i = δi;j. figurations where the lines through each point have the configuration V . One ramified, and everything is reduced to the same behavior on the Prym map. αj R 1 2 This gives us a matrix whose entries are !i called the period matrix Ω. This example is that if we start with P (k) we get P (k): βj matrix is well-defined up to symplectic change of basis and is an invariant of C. 6 Contracted Loci Theorem (Torelli Theorem). The map associating to each curve C the orbit of its • • period matrix Ω is injective. In [Don81], Donagi described the tetragonal construction, which takes a tower of • +3 •• • a double cover of a tetragonal curve to two more such towers, and is a triality: The Schottky problem asks for conditions on which a potential period matrix is • ••• actually the period matrix of a curve, most formally, it asks for equations for the ~ locus of Jacobians among all abelian varieties. C1 Similarly, if we start with two points that are not colinear, the configuration we get is a space quadric, with each point lying on exactly two lines: C1 2 Notation C~ P1 ( • M moduli space of genus g curves ~ g C S •• C2 • | Ag moduli space of g-dimensional ppav’s 1 C2 Jg : Mg !Ag The Torelli map, and also the image of the map P RMg f(C; µ)jC 2 Mg; µ a nontrivial point of order 2 on Jg(C)g 1 0 P Pg : RMg !Ag−1 The Prym map ker (Nm) RAg f(A; µ)jA 2 Ag; µ a nontrivial point of order 2 on Ag But also, all three double covers give the same Prym variety, and so between RJ g : RMg ! RAg the lift of Jg to RMg, and its image that and a theorem of Mumford, we can determine all of the fibers of the Prym RC0 Intermediate Jacobians of cubic 3folds with a marked map. even point of order 2 Theorem ([Mum74]). Let C be a smooth non-hyperelliptic curve of genus 5. If 1 there exists a positive dimensional family of g4’s on C, we must have that C is 3 Theta Functions trigonal, bielliptic or a plane quintic. We want to find equations on A for the closure of the image of J . The natural And a detailed analysis along with the above tells us that the only locus in RS 5 5 And a significant example which occurs in studying the Prym map P : RM ! 5 coordinates to do this with live on , the Siegel upper half space of g × g sym- 6 6 that β contracts is RA × A . Hg A , which uses the tetragonal construction, if we start with five points, the twenty- 5 1 4 metric matrices with positive definite imaginary part. If , δ 2 ( = )g, we have 5 Q Z seven lines on the cubic surface is such a configuration: theta functions 7 Schottky-Jung Relations X θ (Ω; z) = exp πi (n + )tΩ(n + ) + 2(n + )t(z + δ) δ 0 g Between the above, and the fact that the local degree of β on RC is 1, RJ 5 is n2Z 54 and @IRA is 64, we have: and these can be used to construct maps: 5 Theorem. 0 I RS5 = RJ 5 [ RC [ @ RA5 [ RA1 × A4 1 g α = θ (2Ω; 0) 2 = 0 2Z Z And this theorem solves the Schottky problem in genus five, giving us that g−1 small 0 1 S5 = J5. β = θ (2Ω; 0) 2 = 0 1=2 2Z Z References These maps α and β define maps from Ag and RAg to a quotient of projective space P, and Schottky and Jung [SJ09] proved that the diagram commutes. [Don81] Ron Donagi. The tetragonal construction. Bull. Amer. Math. Soc. (N.S.), The most significant example comes from the fact that we can obtain a natural 4(2):181–185, 1981. configuration on the fibers of β , such that each point is on 27 lines, configured RM 5 [Don87] Ron Donagi. Big Schottky. Invent. Math., 89(3):569–599, 1987. g like the lines on a cubic surface, and the same construction gives us the config- Pg RJ g [Iza91] Elham Izadi. On the moduli space of four dimensional principally polar- uration of points and lines on the quadric Ag−1 RAg ized abelian varieties. PhD thesis, Univ. of Utah, 1991. α 2 2 7 [Mum74] David Mumford. Prym varieties. I. In Contributions to analysis (a collec- β V (x1 + x1x2 + x2 + x3x4 + x5x6 + x7x8) ⊂ P (F2) tion of papers dedicated to Lipman Bers), pages 325–350. Academic P Press, New York, 1974. −1 They conjectured that RSg = β (α(Ag−1)) is precisely RJ g, the locus of Jaco- This tells us that the fibers of β5, generically, are unions of this incidence, and so [SJ09] F. Schottky and H. Jung. Neue satze¨ uber¨ symmetralfunctionen und die bians in RAg. This is not quite correct, and we work out a description of RS5. is a multiple of 119. abelschen funktionen. S.-B. Berlin Akad. Wiss., 1909..