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 study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.
We present a treatment of the algebraic description of the Jacobian of a generic genus two plane curve which exploits an SL2(k) equivariance and clarifes the structure of E.V.Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety.
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.
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.
In this article we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of $\Sym^m C$ into $\Sym^n C$ for $m\leq n$, where $C$ is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self product of theta divisor into the self product of the…
We prove that the geometric genus p of a curve in a very generic Jacobian of dimension g>3 satisfies either p=g or p>2g-3. This gives a positive answer to a conjecture of Naranjo and Pirola. For low values of g the second inequality can be further improved to p>2g-2.
The aim of this note is to describe the restriction map from the moduli space of stable rank 2 bundle with small $c_2$ on a jacobian $X$ of dimension 2, to the moduli space of stable rank 2 bundles on the corresponding genus 2 curve $C$ embedded in $X$.
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.
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 prove that in most cases relevant to cryptography, the Frobenius endomorphism on the Jacobian of a genus two curve is represented by a diagonal matrix with respect to an appropriate basis of the subgroup of l-torsion points. From this fact we get an explicit description of the Weil-pairing on the subgroup of l-torsion points. Finally, the explicit description of the Weil-pairing provides us with an efficient, probabilistic algorithm to find generators of the subgroup of l-torsion points on the Jacobian of a genus two curve.
Given a canonical genus three curve $X=\{F=0\}$, we construct, emulating Mumford discussion for hyperelliptic curves, a set of equations for an affine open subset of the jacobian $JX$. We give explicit algorithms describing the law group in $JX$. Finally we introduce a related construction by means of an imbedding of the open set previously described in a Grassmanian variety.
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the Abel-Jacobi, Riemann-Roch and Riemann theta divisor theorems.
For a nonsingular projective curve $C$ of genus 3 defined over an algebraically closed field of characteristic $p > 2$, we give a necessary and sufficient condition that the Jacobian variety $J(C)$ has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.
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
L. Moret-Bailly constructed families $\mathfrak{C}\rightarrow \mathbb{P}^1$ of genus 2 curves with supersingular jacobian. In this paper we first classify the reducible fibers of a Moret-Bailly family using linear algebra over a quaternion algebra. The main result is an algorithm that exploits properties of two reducible fibers to compute a hyperelliptic model for any irreducible fiber of a Moret-Bailly family.
We prove that k-th singular cohomology group of the complement of the theta divisor in a hyperelliptic Jacobian is isomorphic to the k-th fundamental representation of the symplectic group Sp(2g,C). This is one of the conjectures in our previous paper.
The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we describe how to extend these families over the moduli space of stable marked curves (or rather an open subset thereof) using a stability parameter. We then prove a wall-crossing formula describing how the theta divisor varies with the stability parameter. We use that result to analyze a divisor on the moduli space of smooth marked curves that has recently be…
We construct an infinite number of Shimura curves contained in the locus of hyperelliptic Jacobians of genus 3. In the opposite direction, we show that in genus 3 the only possible non-complete (in the moduli space of abelian threefolds) Kuga curves contained in the hyperelliptic locus have the same degeneration data as that of the examples we construct. The locus of genus 3 hyperelliptic Jacobians is a divisor within the moduli space of principally polarized abelian threefolds, and our result demonstrates the techniques we develop more generally for dealing with Shimura curves contained withi…
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.
A set of multi-homogeneous equations for the Jacobian of a genus two curve is given. The approach used is to write down affine equations for the Jacobian minus various tranlations of the Theta-divisor by [2]-division points, and then to write down affine glue equations for the overlaps. Taking multi-projective completions for all of these then yields a complete multi-projective two-dimensional variety whose points are in one-to-one correspondence with degree zero divisor classes on the curve (i.e. the Picard group). This multi-projective variety then becomes a complete projective two-dimension…
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on $Pic^{g-1}C$ which are linearly equivalent to $2Θ$. The embedded tangent space at a semi-stable non-stable bundle $ξ\oplusξ^{-1}$, where $ξ$ is a degree zero line bundle, is shown to consist of those divisors in $|2Θ|$ which contain $Sing(Θ_ξ)$ where $Θ_ξ$ is the translate of $Θ$ by $ξ$. We also obtain geometrical results on the structure of this tangent space.
We show that the only summands of theta divisors on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the obvious powers of the curve and the Fano surface of lines on the threefold. The proof uses the decomposition theorem for perverse sheaves, a bit of representation theory and a computation of characteristic cycles for Brill-Noether sheaves.
Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves with complex multiplication. In particular, we show that the Weil- and the Tate-pairing on such a Jacobian are non-degenerate over the same field extension of the ground field.
We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along t…
Let E be an elliptic curve defined over a finite field. Balasubramanian and Koblitz have proved that if the l-th roots of unity m_l is not contained in the ground field, then a field extension of the ground field contains m_l if and only if the l-torsion points of E are rational over the same field extension. We generalize this result to Jacobians of genus two curves. In particular, we show that the Weil- and the Tate-pairing are non-degenerate over the same field extension of the ground field. From this generalization we get a complete description of the l-torsion subgroups of Jacobians of su…
For each universal genus-$g$ polarization $μ$ of degree $d$, we construct a universal tropical Jacobian $J_{μ,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several properties of the space $J_{μ,g}^{trop}$. In particular, we prove that the natural compactification of $J_{μ,g}^{trop}$ is the tropicalization of the Esteves' compactified universal Jacobian over the moduli space of stable pointed genus-$g$ curves.
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac} (X))\otimes Q$ contains the totally real cubic number field $Q(ζ_7 +\barζ_7 )$. We construct explicit three-dimensional families whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when $X$ is hyperelliptic was studied in a previous work by Hoffman and Wang and some nonhyperelliptic curves were constructed in a previous paper by Hoffman, Z. Liang. Sakai and Wang.
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.
In this work we consider constructions of genus three curves $X$ such that $\mathrm{End}(\mathrm{Jac}(X)) \otimes Q$ contains the totally real cubic number field $Q(ζ_ 7 + \overlineζ_7)$. We construct explicit two-dimensional families defined over $Q(s; t)$ whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied by the authors Hoffman and Wang in a previous work. We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form.
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.
The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a canonical divisor $Θ$, called the Theta divisor. It is well-known that $Θ$ is ample and, in fact, $3Θ$ is very ample. For a general complete, integral curve $X$, D'Souza constructed a compactification $\bar J$ of the Jacobian $J$ by considering torsion-free, rank 1 sheaves on $X$. Soucaris and the author considered independently the analogous Theta divisor $Θ$ on $\bar J$, and showed that $Θ$ is ample. In this article, we show that $nΘ$ is very ample for $n$ greater or equal to a specified lower bound. If $X$ has at most ordi…
We study the singular locus on the algebraic surface $§_n$ of genus 2 curves with a $(n, n)$-split Jacobian. Such surface was computed by Shaska in \cite{deg3} for $n=3$, and Shaska at al. in \cite{deg5} for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface $§_3$ discovered in \cite{deg3} to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We furt…
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.
We give a closed formula for recovering a non-hyperelliptic genus three curve from its period matrix, and derive some identities between Jacobian Nullwerte in dimension three.
We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli $\mathcal H_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak p \in \mathcal H_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal X$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_…
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).
In this paper we establish that the singularities of the universal compactified Jacobian are canonical if the genus is at least four. As a corollary we determine the Kodaira dimension and the Iitaka fibration of the universal compactified Jacobian for every degree and genus. We also determine the birational automorphism group for every degree if the genus is at least twelve. This extends work of G. Farkas and A. Verra, as well as that of G. Bini, C. Fontanari and the third author.
We give an algorithm to compute $(\ell,\ell,\ell)$-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.
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 …
In this paper we study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure (IVHS) is of rank 1. We show that if the genus g is greater or equal to 8 or g=6,7 and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete a result of Naranjo and Pirola. We show in fact that if the genus g is greater or equal to 6, the hyperelliptic locus is the only 2g-1-dimensional sub-locus Y of the moduli space of curves of genus g, such that for the general element [C] in Y, its Ja…
Let $C$ be a curve of genus 2 and $ψ_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $ϕ_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering. We determine all possible ramifications for $ϕ_1$. If $ψ_1:C \lar E_1$ is maximal then there exists a maximal map $ψ_2:C\lar E_2$, of degree $n$, to some elliptic curve $E_2$ such that there is an isogeny of degree $n^2$ from the Jacobian $J_C$ to $E_1 \times E_2$. We say that $J_C$ is $(n,n)$-decomposable. If the degree $n$ is odd the pair $(ψ_2, E_2)$ is canonic…
We prove the following converse of Riemann's Theorem: let (A,Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety Θ=C+Y. Then C is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As applications, we determine all theta divisors that are dominated by a product of curves and characterize Jacobians by the existence of a d-dimensional subvariety with curve summand whose twisted ideal sheaf is a generic vanishing sheaf.
In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers, hyperelliptic curve cryptography, etc. Choosing genus 2 curves suitable for such applications is an important step of such algorithms. In existing algorithms often such curves are chosen using equations of moduli spaces of curves with decomposable Jacobians or Humbert surfaces. In these lectures we will cover basic properties of genus 2 curves, moduli spaces…
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…
We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding addition map, and show that the minimum can only be achieved if X is a theta divisor. Conjecturally, the latter happens only on Jacobians of curves and intermediate Jacobians of cubic threefolds. As an application, we prove that nondegenerate generic vanishing subschemes of indecomposable principally polarized abelian varieties are automatically reduced and irreduc…
Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genu…
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…
Let $p$ be an odd prime number and be an integer coprime to $p$. We survey an algorithm for computing explicit rational representations of $(\ell,...,\ell)$-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension $K$ of the field of $p$-adic numbers $\mathbb{Q}\_p$. The algorithm has a quasi-linear complexity in $\ell$ as well as in the genus of the curves.
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.
