Effective reconstruction formulas of a curve from its theta hyperplanes are known classically in genus 2 (where the theta hyperplanes are Weierstrass points), and 3 (where, for a generic curve, the theta hyperplanes are bitangents to a plane quartic). However, for higher genera, no formula or algorithm are known. In this paper we give an explicit (and simple) algorithm for computing a generic genus 4 curve from it's theta hyperplanes.
We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.
Let C be a curve of genus g, and let SU(r) be the moduli space of vector bundles of rank r on C, with trivial determinant. A general E in SU(r) defines a theta divisor in the linear system |r Theta|, where Theta is the canonical theta divisor in Pic^{g-1}(C). This defines a rational map SU(r) - - > |r Theta|, which is the map associated to the determinant bundle on SU(r) (the positive generator of Pic(SU(r)). In this paper we prove that in genus 2 this map is generically finite and dominant. The same method, together with some classical work of Morin, shows that in rank 3 and genus 3 the theta…
We discuss variations of Hodge structures on abelian varieties that arise from intersecting translates of theta divisors with a special focus on the case of abelian varieties of dimension 4
In this paper we prove a conjecture of Hershel Farkas that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian. We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.
We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli …
We study the Albanese image of a compact Kähler manifold whose geometric genus is one. We prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.
Let U(r) be the moduli space of rank r vector bundles with trivial determinant on a smooth curve of genus 2. The map theta_r: U(r) -> |r Theta|, which associates to a general bundle its theta divisor, is generically finite. In this paper we give a geometric interpretation of the generic fibre of theta_r.
In this paper we investigate generalized theta divisors $Θ_r$ in the moduli spaces $\mathcal{U}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $Φ$ of $Θ_r$ in terms of a projective bundle $π:\mathbb{P}(\mathcal{V})\to\mathcal{U}_C(r-1,r)$ which parametrizes extensions of stable vector bundles on the base by $\mathcal{O}_C$. Then, we study the composition of $Φ$ with the well known theta map $θ$. We prove that, when it is restricted to the general fiber of $π$, we obtain a linear embedding.
According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve G, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of G. It is proven that, for a generic X, the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c_1=-K_X and deg c_2=6 and that the latter is birational to the singular locus of the theta divisor of J(X).
We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $Θ$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $Θ$.
Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ even theta constants define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)=H_g/Γ_g[4,8]$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$X_3(4,8)\to P^{35}$$ is biholomorphic onto the …
Li-Zinger's hyperplane theorem states that the genus one GW-invariants of the quintic threefold is the sum of its reduced genus one GW-invariants and 1/12 multiplies of its genus zero GW-invariants. We apply the Guffin-Sharpe-Witten's theory (GSW theory) to give an algebro-geometric proof of the hyperplane theorem, including separation of contributions and computation of 1/12.
We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.
The moduli space M_2 of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 = 1-g over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Ray…
A generic strictly semistable bundle of degree zero over a curve X has a reducible theta divisor, given by the sum of the theta divisors of the stable summands of the associated graded bundle. The converse is not true: Beauville and Raynaud have each constructed stable bundles with reducible theta divisors. For X of genus at least 5, we construct stable vector bundles over X of rank $r$ for all $r \geq 5$, with reducible and nonreduced theta divisors. We also adapt the construction to symplectic bundles. In the appendix, Raynaud's original example of a stable rank 2 vector bundle with reducibl…
In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.
The level moduli space $A_g^{4,8}$ is mapped to the projective space by means of gradients of odd Theta functions, such a map turning out no to be injective in the genus 2 case. In this work a congruence subgroup $Γ$ is located between $Γ_2(4,8)$ and $Γ_2(2,4)$ in such a way the map factors on the related level moduli space $A_Γ$, the new map being injective on $A_Γ$. Satake's compactification $\text{Proj}A(Γ)$ and the desingularization $\text{Proj}S(Γ)$ are also due to be investigated, since the map does not extend to the boundary of the compactification; to aim at this, an algebraic descript…
We give explicit formulas for the Kawazumi-Zhang invariant and Faltings delta-invariant of a compact and connected Riemann surface of genus three. The formulas are in terms of two integrals over the associated jacobian, one integral involving the standard Riemann theta function, and another involving a theta function particular to genus three that was discovered by Frobenius. We review part of Frobenius' work on his theta function and connect our results with a formula due to Bloch, Hain and Bost describing the archimedean height pairing of Ceresa cycles in genus three.
In this paper we compute certain two-point integrals over a moduli space of stable maps into projective space. Computation of one-point analogues of these integrals constitutes a proof of mirror symmetry for genus-zero one-point Gromov-Witten invariants of projective hypersurfaces. The integrals computed in this paper constitute a significant portion in the proof of mirror symmetry for genus-one GW-invariants completed in a separate paper. These integrals also provide explicit mirror formulas for genus-zero two-point GW-invariants of projective hypersurfaces. The approach described in this pap…
Raynaud has shown that over a general curve of genus $g \ge 2$, every semistable bundle of rank three and integral slope admits a theta divisor. We show that this can fail for special curves: Over any bielliptic curve of genus $g \ge 5$, we construct a stable rank three bundle of trivial determinant with no theta divisor. This gives a partial answer to a question of Beauville.
We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.
We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus two square-tiled surfaces with these invariants. The main technical tool is the computation of divisor classes of Hilbert Jacobi forms on the universal abelian surface over the pseudo-Hilbert modular surface.
In this note we prove the genus 3 case of a conjecture of G. Farkas and A. Verra on the limit of the Scorza correspondence for curves with a theta-null. Specifically, we show that the limit of the Scorza correspondence for a hyperelliptic genus 3 curve C is the union of the curve ${x,σ(x))$ (where $σ$ is the hyperelliptic involution), and twice the diagonal. Our proof uses the geometry of the subsystem Γ_{00} of the linear system 2Θ, and Riemann identities for theta constants.
Our main result is an effective version of the Torelli theorem in genus $3$ and any characteristic not $2$: the configuration of the odd theta characteristics of a curve $C$ of genus $3$ determines a del Pezzo surface $S$ of degree two and that $C$ is recovered as the normalization of a certain curve on $S$. This surface is not the one usually associated to $C$; we describe the distinction. We also give an explicit geometrical description of the quadratic twist (observed by Serre) that arises in the statement of the Torelli theorem.
We compute the rational cohomology of the moduli space of non-singular non-hyperelliptic complex projective curves of genus 3 with an odd theta characteristic.
In this paper we study the sections of the canonical line bundle on the moduli space of parabolic semistable vector bundles with trivial determinant and fixed parabolic structure of type $\underlineλ=(λ_1,..., λ_s)$ (with each weight $λ_i$ in $P_{\ell}(\SL(r))$) on a smooth projective irreducible curve over $\C$ of genus $g \geq 1$. We give a nontrivial lower bound for the dimension of the sections (that are called generalized parabolic SL(r)-theta functions) when $\sum_{1}^{s} λ_i$ is in the root lattice.
An effective algorithm of determining Gromov--Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound) from Gromov--Witten invariants of the ambient space is proposed. The Appendix is joint with E. Schulte-Geers.
We give an explicit description of theta-characteristics on tropical curves and characterize the effective ones. We construct the moduli space for tropical theta-characteristics of genus g as a generalized cone complex. We describe the fibers of the specialization map from the classical moduli space of theta-characteristics on algebraic curves to our moduli space of tropical theta-characteristics.
Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.
We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the Andreotti-Mayer loci we study subvarieties of principally polarized abelian varieties (B,Theta) parametrizing points b such that Theta and the translate Theta_b are tangentially degenerate along a variety of a given dimension.
Let $\X$ be an irreducible, smooth, projective curve of genus $g \geq 2$ defined over the complex field $\C.$ Then there is a covering $π: \X \longrightarrow ¶^1,$ where $¶^1$ denotes the projective line. The problem of expressing branch points of the covering $π$ in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among…
We give a criterion to distinguish between a genus three Jacobian and its [-1] twist in terms of the product of the 36 even theta nulls. We also express the product of the 36 theta nulls in terms of the discriminant of a genus three curve. The results are arithmetic in nature and thus add to previous work over C on the product of the even theta nulls. They generalize previous work of Ritzenthaler and Lachaud for Abelian threefolds which are (2,2,2) isogenous to a product of elliptic curves. This answers a 2003 of letter of Jean-Pierre Serre to Jaap Top.
Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let I_C be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that, if g\geq 8, and if the Clifford index of C is greater than 2, then H^1(P^{g-1}, I_C^2(k))=0, for k\geq 3. We prove this conjecture for g\geq 11. The second conjecture states that a Brill-Noether-Petri curve of genus g\geq 12 …
We analyze the stratification of the moduli space S_g of spin curves of genus g given by the dimension of the theta-characteristic. Using the relation between gaussian maps and the strata S_g^r, we construct "regular" components of S_g^r having expected codimension r(r+1)/2 inside S_g. We also relate moduli spaces of pointed curves with a moving spin structure to the classical Gieseker-Petri loci in M_g. We show that the locus of curves for which the Gieseker-Petri theorem fails for a pencil is always a divisor on M_g. Finally, we give a sufficient criterion for the injectivity of Gaussian map…
S. Kondō used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by: hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational "splitting curves".
In the paper "The universal Kummer threefold", Q. Ren, S. Sam, G. Schrader, and B. Sturmfels (arXiv:1208.1229), conjectured equations for the universal Kummer variety in genus 3 case. Though, most of these equations are obtained from the Fourier-Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic was obtained differently. The aim of the current paper is to show that Coble's quartic can be obtained as Fourier-Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in "…
We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let $J$ be the Jacobian of a curve $X$, and let $W_d \subset J$ be the locus of effective divisor classes of degree $d$. We show that the pair $(J^{an},W_d^{an})$ is $d$-connected, and thus in particular the inclusion of the analytification of the theta divisor $Θ^{an}$ into $J^{an}$ satisfies a Lefschetz hyperplane theorem for $\mathbb{Z}$-cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrar…
By a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (greater than 280) of a general polarized K3 surface, then the second gaussian map of C is surjective. The resulting bound for the genus g of a general curve with surjective second gaussian map is decreased to g >152.
We discuss Weber's formula which gives the quotient of two Thetanullwerte for a plane smooth quartic in terms of the bitangents. In particular, we show how it can easily be derived from the Riemann-Jacobi formula.
Let C be a curve of genus g and L a line bundle of degree 2g on C . Let M be the kernel of the evaluation map from the trivial bundle with fibre H^0(C,L) into L . We show that when L is general enough, the rank g bundle M and its exterior powers are stable and admit a reducible theta divisor.
A general canonical curve X determines a finite set T(X) of hyperplanes, which is in bijective correspondence with the set of odd theta-characteristics of X. The definition of T(X) can be extended to certain singular curves, in a way that is compatible with degenerations. If X is a Deligne-Mumford stable curve (or a cuspidal curve), we describe the enumerative and local geometry of T(X); this is applied to show that some singular curves are uniquely determined by T(X). This work generalizes the preliminary steps used in AG/0008239 (with E. Sernesi) to show that a general curve of genus 3 can b…
The Kawazumi-Zhang invariant $φ$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $φ$ in the non-separating degeneration limit, it is shown that $φ$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $φ$ near the non-separating node, gives full control on the asymptotics of $φ$ in the various degeneration limits, and p…
We show that a general prime Fano threefold X of genus 8 can be reconstructed from the pair $(Γ,L)$, where $Γ$ is its Fano curve of lines and $L=O_Γ(1)$ is the theta-characteristic which gives a natural embedding $Γ\subset \matbb{P}^5$.
It is well known that, fixed an even, unimodular, positive definite quadratic form, one can construct a modular form in each genus; this form is called the theta series associated to the quadratic form. Varying the quadratic form, one obtains the ring of stable modular forms. We show that the differences of theta series associated to specific pairs of quadratic forms vanish on the locus of hyperelliptic Jacobians in each genus. In our examples, the quadratic forms have rank 24, 32 and 48. The proof relies on a geometric result about the boundary of the Satake compactification of the hyperellip…
The aim of this note is to give a precise description of the local structure of the moduli space of rank 3 vector bundles over a curve of genus 2, which is in particular shown to be a local complete intersection. This allows us to investigate the local structure of the branch locus of the theta map, whose dual is known to be the Coble cubic in P^8.
Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $Θ_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r Θ|$ is generically injective for large values of $g$.
We give an algebraic method to compute the fourth power of the quotient of any even theta constants associated to a given non-hyperelliptic curve in terms of geometry of the curve. In order to apply the method, we work out non-hyperelliptic curves of genus 4, in particular, such curves lying on a singular quadric, which arise from del Pezzo surfaces of degree 1. Indeed, we obtain a complete 2-level structure of the curves by studying their theta characteristic divisors via exceptional divisors of the del Pezzo surfaces as the structure is required for the method.
Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the determinant line bundle $\cL$ over $\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\cB_{g+2}$ over any curve $C$, and base points in $\cB_4$ over any hyperelliptic curve.
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Vélu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
Let p:C' -> C be an unramified double covering of irreducible smooth curves and let P be the attached Prym variety. We prove the schematic theta-dual equalities in the Prym variety T(C')=V^2 and T(V^2)=C', where V^2 is the Brill-Noether locus of P associated to p considered by Welters. As an application we prove a Torelli Theorem analogous to the fact that the g-th symmetric product of a curve D of genus g determines the curve.
Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may ask whether the corresponding linear series contains an effective divisor with totally real support. This translates into a particular type of parametrized real root counting problem that we wish to solve exactly. On the other hand, it is known that for a given genus and number of real connected components, any linear series of sufficiently large degree contai…
Theta-duality and Prym-Torelli for cyclic covers of hyperelliptic curves
Anatoli Shatsila Doctoral School of Exact；Natural Sciences, Jagiellonian University in Krak\'ow, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland
Let $P \cup P'$ be the two component Prym variety associated to an étale double cover $\tilde{C} \to C$ of a non-hyperelliptic curve of genus $g \geq 6$ and let $|2Ξ_0|$ and $|2Ξ_0'|$ be the linear systems of second order theta divisors on $P$ and $P'$ respectively. The component $P'$ contains canonically the Prym curve $\tilde{C}$. We show that the base locus of the subseries of divisors containing $\tilde{C} \subset P'$ is scheme-theoretically the curve $\tilde{C}$. We also prove canonical isomorphisms between some subseries of $|2Ξ_0|$ and $|2Ξ_0'|$ and some subseries of second order theta …
