Theta Divisor Research Articles

Raynaud’s vector bundles and base points of the generalized Theta divisor

We study base points of the generalized Θ-divisor on the moduli space of vector bundles on a smooth algebraic curve X of genus g defined over an algebraically closed field. To do so, we use the derived categories Db(Pic0(X)) and Db(Jac(X)) and the equivalence between them given by the Fourier–Mukai transform \({\rm FM}_\mathcal {P}\) coming from the Poincaré bundle. The vector bundles P m on the curve X defined by Raynaud play a central role in this description. Indeed, we show that E is a base point of the generalized Θ-divisor, if and only if there exists a non-trivial homomorphism Prk(E)g+1 → E.

On Witt vector cohomology for singular varieties

Over a perfect field $k$ of characteristic $p > 0$, we construct a vector cohomology with compact for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.

Hauteur de Faltings et hauteur de Néron–Tate du diviseur thêta

In this paper we prove a formula relating the Faltings height of an abelian variety $A$ over $\bar{\mathbb{Q}}$ and the Neron–Tate height of a theta divisor on $A$ .

Jacobian variety and integrable system — after Mumford, Beauville and Vanhaecke

Beauville [A. Beauville, Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables, Acta. Math. 164 (1990) 211–235] introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system [D. Mumford, Tata Lectures on Theta II, Birkhäuser, 1984]. In this article, we construct a variant of Beauville’s system whose general level set is isomorphic to the complement of the intersection of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke [P. Vanhaecke, Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z. 211 (1992) 265–313; P. Vanhaecke, Integrable systems in the realm of algebraic geometry, in: Lecture Notes in Mathematics, vol. 1638, 2001].

Vector bundles and theta functions on curves of genus 2 and 3

Let SU C (r) be the moduli space of vector bundles of rank r and trivial determinant on a curve C . A general E in SU C (r) defines a divisor Θ E in the linear system | r Θ|, where Θ is the canonical theta divisor in Pic g -1 ( C ). This defines a rational map Θ: SU C (r) → | r Θ|, which turns out to be the map associated to the determinant bundle on SU C (r) (the positive generator of Pic ( SU C (r) ). In genus 2 we prove that 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 map is a finite morphism - in other words, every vector bundle in SU C (3) admits a theta divisor.

Systems of Hess-Appel’rot Type

We construct higher-dimensional generalizations of the classical Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of this new class of systems, which is closely related to the integration of the Lagrange bitop performed by us recently and uses Mumford relation for theta divisors of double unramified coverings. Based on the basic properties satisfied by such a class of systems related to bi-Poisson structure, quasi-homogeneity, and conditions on the Kowalevski exponents, we suggest an axiomatic approach leading to what we call the "class of systems of Hess-Appel'rot type".

Genus 4 trigonal reduction of the Benney equations

It was shown by Gibbons and Tsarev (1996 Phys. Lett. A 211 19, 1999 Phys. Lett. A 258 263) that N-parameter reductions of the Benney equations correspond to particular N-parameter families of conformal maps. In recent papers (Baldwin and Gibbons 2003 J. Phys. A: Math. Gen. 36 8393–417, Baldwin and Gibbons 2004 J. Phys. A: Math. Gen. 37 5341–54), the present authors have constructed examples of such reductions where the mappings take the upper half p-plane to a polygonal slit domain in the λ-plane. In those cases, the mapping function was expressed in terms of the derivatives of Kleinian σ functions of hyperelliptic curves, restricted to the one-dimensional stratum Θ1 of the Θ-divisor. This was done using an extension of the method given in Enolskii et al (2003 J. Nonlinear Sci. 13 157) extended to a genus 3 curve (Enolski V Z and Gibbons J Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, in preparation). Here, we use similar ideas, but now applied to a trigonal curve of genus 4. Fundamental to this approach is a family of differential relations which σ satisfies on the divisor. Again, it is shown that the mapping function is expressible in terms of quotients of derivatives of σ on the divisor Θ1. One significant by-product is an expansion of the leading terms of the Taylor series of σ for the given family of (3, 5) curves; to the best of the authors' knowledge, this is new.

Fourier-Mukai Transform and Adiabatic Curvature of Spectral Bundles for Landau Hamiltonians on Riemann Surfaces

We study the family of Landau Hamiltonians on a Riemann surface S by means of a Nahm transform and an integral functor related to the Fourier-Mukai transform associated to its jacobian variety J(S). This approach allows us to explicitly determine the spectral bundles associated to the holomorphic Landau levels. As a first main result we prove that these spectral bundles are holomorphic stable bundles with respect to the canonical polarization of J(S) determined by the theta divisor . The spectral bundles are endowed with a natural connection and the adiabatic charge transport properties of the corresponding Landau levels are determined by the adiabatic curvature of , which coincides with the curvature of the determinant bundle det . By means of the theory of analytic torsion and determinant bundles developed by Bismut, Gillet and Soule we compute the adiabatic curvature of the spectral bundles on an arbitrary Riemann surface. We prove that all the holomorphic Landau levels have the same charge transport coefficients but their adiabatic curvatures differ by a term which involves the relative analytic torsion of different powers of the canonical bundle of S twisted by a fixed line bundle.

Efficient Hyperelliptic Curve Cryptosystems Using Theta Divisors

It has recently been reported that the performance of hyperelliptic curve cryptosystems (HECC) is competitive to that of elliptic curve cryptosystems (ECC). Concerning the security of HECC, the theta divisors play an important role. The scalar multiplication using a random base point is vulnerable to an exceptional procedure attack, which is a kind of side-channel attacks, using theta divisors. In the case of cryptographic protocols of the scalar multiplication using fixed base point, however, the exceptional procedure attack is not applicable. First, we present novel efficient scalar multiplication using theta divisors, which is the positive application of theta divisors on HECC. Second, we develop a window-based method using theta divisors that is secure against side-channel attacks. It is not obvious how to construct a base point D such that all pre-computed points are theta divisors. We present an explicit algorithm for generating such divisors.

The theta divisor and the Stickelberger theorem

This paper is devoted to studying certain trivial correspondences provided by theta divisors and their relation to the Brumer-Stark conjecture. © 2005 American Mathematical Society.

A lower bound for the dimension of the base locus of the generalized theta divisor

We produce a lower bound for the dimension of the base locus of the generalized theta divisor Θ r on the moduli space SU C ( r ) of semistable vector bundles of rank r and trivial determinant on a smooth curve C of genus g ⩾ 2 . To cite this article: D. Arcara, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

A theta divisor containing an abelian subvariety

We construct a Jacobian of dimension three whose theta divisor contains an elliptic curve. We work over an algebraically closed .eld of characteristic zero.

Torsion on theta divisors of hyperelliptic Fermat Jacobians

We generalize a result of Anderson by showing that torsion points of certain orders cannot lie on a theta divisor in the Jacobians of hyperelliptic images of Fermat curves. The proofs utilize the explicit geometry of hyperelliptic Jacobians and their formal groups at the origin.

Cubic threefolds and abelian varieties of dimension five

This paper proves the following converse to a theorem of Mumford: Let A A be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then A A is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of A A , and ultimately to show that A A is the Prym variety of a possibly singular plane quintic.

Effective very ampleness for generalized theta divisors

Given a smooth projective curve X, we give effective very ampleness bounds for generalized theta divisors on the moduli spaces SUX(r,d) and UX(r,d) of semistable vector bundles of rank r and degree d on X with fixed, respectively, arbitrary, determinant.

A necessary and sufficient condition for Riemann's singularity theorem to hold on a Prym theta divisor

Let $(P,\Xi)$ be the naturally polarized model of the Prym variety associated to the etale double cover $\pi : \tilde C\rightarrow C$ of smooth connected curves defined over an algebraically closed field k of characteristic $\ne 2$ , where genus( C ) = $g \ge 3$ , Pic $^{(2g-2)}(\tilde C) \supset P = \{\mathcal L \in {\rm Pic}^{(2g-2)}(\tilde C) : {\rm Nm}(\mathcal L) = \omega_C$ and $h^0(\tilde C,\mathcal L)$ is even\} is the Prym variety, and $P \supset \Xi = \{\mathcal L \in P: h^0(\tilde C,\mathcal L) >0 \}$ is the Prym theta divisor with its reduced scheme structure. If $\mathcal L$ is any point on $\Xi$ , we prove that ‘Riemann's singularity theorem holds at $\mathcal L$ ’, i.e. mult $_{\mathcal L}(\Xi) = (1/2)h^0(\tilde C,\mathcal L)$ , if and only if $\mathcal L$ cannot be expressed as $\pi^*(\mathcal M)(B)$ where $B \ge 0$ is an effective divisor on $\tilde C$ , and $\mathcal M$ is a line bundle on C with $h^0(C,\mathcal M) >(1/2)h^0(\tilde C,\mathcal L)$ . This completely characterizes points of $\Xi$ where the tangent cone is the set theoretic restriction of the tangent cone of $\tilde {\Theta}$ , hence also those points on $\Xi$ where Mumford's Pfaffian equation defines the tangent cone to $\Xi$ .

Divisors on [formula omitted] and the minimal resolution conjecture for points on canonical curves

We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles.

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in P 3g−3 for generic curves), described by 2 Θ functions or second logarithmic derivatives of theta. We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.

Delzant models of moduli spaces

For every genus we construct a smooth, complete, rational polarized algebraic variety together with an effective normal crossing divisor such that for every moduli space of semistable topologically trivial vector bundles of rank 2 on an algebraic curve of genus there is a holomorphic isomorphism , where is the Kummer variety of the Jacobian of , sending the polarization of to the theta divisor of the moduli space. This isomorphism induces isomorphisms of the spaces and .

Some stable vector bundles with reducible theta divisor

Let C be a curve of genus g and L a line bundle of degree 2g on C. Let ML be the kernel of the evaluation map \(\). We show that when L is general enough, the rank g bundle ML and its exterior powers are stable, but admit a reducible theta divisor.