A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the Andreotti-Mayer approach to the Schottky problem, a result which also reproduces the only previously known case g=4. This new approach, based on the combinatorics of determinantal relations for two-fold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g=2 and g=3. Finally, the induced Siegel volume form is expressed in terms of the Mumford form.
We present a new class of examples of base points for the generalized theta divisor on the moduli space of semistable vector bundles of trivial determinant on a compact Riemann surface and we prove that for sufficiently large rank the base locus is positive dimensional.
Let ϕ: \mathbb{P}^{r}\dashrightarrow Z be a birational transformation with a smooth connected base locus scheme, where Z\subseteq\mathbb{P}^{r+c} is a nondegenerate prime Fano manifold. We call ϕa quadro-quadric special briational transformation if ϕand ϕ^{-1} are defined by linear subsystems of |\mathcal{O}_{\mathbb{P}^{r}}(2)| and |\mathcal{O}_{Z}(2)| respectively. In this paper we classify quadro-quadric special birational transformations in the cases where either (i) Z is a complete intersection and the base locus scheme of ϕ^{-1} is smooth, or (ii) Z is a hypersurface.
We produce a lower bound for the dimension of the base locus of the generalized theta divisor 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 > 1.
We describe explicitly all hyperelliptic limits of quadrics through smooth canonical curves of genus $g$ in ${\mathbb P}^{g-1}$. Also, we construct an open embedding of the blow up of a ${\rm PGL}_g$-bundle over the moduli space of curves of genus $g$ along the hyperelliptic locus into the blow up of the canonical Hilbert scheme of ${\mathbb P}^{g-1}$ along the closure of the locus of canonical ribbons, which are certain double thickenings of rational normal curves introduced and studied by Bayer and Eisenbud.
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics…
We consider two varieties associated to a web of quadrics W in the projective space of dimension 7. One is the base locus and the second one is the double cover of the three dimensional projective space branched along the determinant surface of W. We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W contains a plane then in the generic case the two varieties are birational.
This paper shows that on the moduli space of semi-stable vector bundles of fixed rank and determinant (of any degree) on a smooth curve of genus at least two, the base locus of the generalized theta divisor is large provided the rank is sufficiently large. It also shows that the base locus is large for positive multiples of the theta divisor. This work extends results already known for the case where the determinant is of degree zero.
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 show that the gradient and the hessian of the Riemann theta function in dimension n can be combined to give a theta function of order n+1 and modular weight (n+5)/2 defined on the theta divisor. It can be seen that the zero locus of this theta function essentially gives the ramification locus of the Gauss map. For jacobians this leads to a description in terms of theta functions and their derivatives of the Weierstrass point locus on the associated Riemann surface.
We study birational transformations P^n--->S \subseteq P^N defined by linear systems of quadrics whose base locus is smooth and irreducible of dimension \leq3 and whose image S is sufficiently regular.
The geometry of the Fano scheme of maximal linear spaces contained in the base locus of a pencil of quadrics has been studied by algebraic geometers when the base field is algebraically closed. In this paper, we work over an arbitrary base field of characteristic not equal to 2 and show how these Fano schemes are related to the Jacobians of hyperelliptic curves. In particular, if $B$ is the base locus of a generic pencil of quadrics in $\bbp^{2n+1}$, and $F$ is the Fano variety of $n - 1$ planes contained in $B$, then $F$ is a component of a disconnected commutative algebraic group $G = \picz(…
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.
In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the theta-null divisor where this singularity is not an ordinary double point. By using theta function methods we first show that this locus does not equal the entire theta-null divisor (this was shown previously by O. Debarre). We then show that this locus is contained in the intersection of the theta-null divisor with the other irreducible components of the And…
A linear system of real quadratic forms defines a real projective variety. The real non-singular locus of this variety (more precisely of the underlying scheme) has a highly connected double cover as long as each non-zero form in the system has sufficiently high Witt index.
Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular forms, defined on the Teichmuller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus 4 is proportional to the square root of the products of Thetanullwerte χ_{68}, which is a proof of the recently rediscovered…
We prove that the set of singular configurations of a general Gough Stewart platform has a rational parametrization. We introduce a reciprocal twist mapping which, for a general orientation of the platform, realizes the cubic surface of singularities as the blowing up of a quadric surface in five points.
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.
Given a vector bundle $E$ of rank $r$ and degree $d$ on a curve $C$ of genus $g$, one can associate to $E$ in a natural way several other vector bundles. For example, one can take wedge powers of $E$. If $E$ is generated by global sections, the kernel of the evaluation map of sections is again a vector bundle. Also, new vector bundles can be produced by taking elementary transformations centered at a fixed point. Under suitable conditions on degree and rank, these constructions can be carried out globally. While all this processes seem quite elementary, very little is known about the resulting…
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.
In an earlier paper we constructed a Cartier divisor on the theta divisor of a principally polarised abelian variety whose support is precisely the ramification locus of the Gauss map. In this note we discuss a Green's function associated to this locus. For jacobians we relate this Green's function to the canonical Green's function of the corresponding Riemann surface.
Let X_R be the zero locus in RP^n of one or two independently and Weyl distributed random real quadratic forms (this is the same as requiring that the corresponding symmetric matrices are in the Gaussian Orthogonal Ensemble). We prove that the sum of the Betti numbers of X_R behaves asymptotically as n (when n goes to infinity). The methods we use combine Random Matrix Theory, Integral geometry and spectral sequences.
We recall that a smooth ample surface $\mathcal{S}$ in a general $(1,2,2)$-polarized abelian threefold, which is the pullback of the Theta divisor of a smooth plane quartic curve $\mathcal{D}$, is a surface isogenous to the product $\mathcal{C} \times \mathcal{C}$, where $\mathcal{C}$ is a genus $9$ curve embedded in $\mathbb{P}^3$ as complete intersection of a smooth quadric and a smooth quartic. We show that the space of global holomorhic sections of the canonical bundle of this surface is generated by certain determinantal bihomogeneous polynomials of bidegree $(2,2)$ on $\mathbb{P}^3$, whi…
Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional proj…
We show that analytic torsion of smooth theta divisor is represented by a Siegel modular form characterizing the Andreotti-Mayer locus if its dimension is positive.
Let $C\subset{\mathbb P}^{g-1}$ be a canonically embedded nonsingular nonhyperelliptic curve of genus $g$ and let $X\subset{\mathbb P}^{g-1}$ be a quadric containing $C$. Our main result states among other things that the Hilbert scheme of $X$ is at $[C\subset X]$ a local complete intersection of dimension $g^2-1$, and is smooth when $X$ is. It also includes the assertion that the minimal obstruction space for this deformation problem is in fact the full associated $\operatorname{Ext}^1$-group and that in particular the deformations of $C$ in $X$ are obstructed in case $C$ meets the singular l…
We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of ${\rm SSP}$ to various loci in the Hilbert scheme of four points in $\mathbb{P}^3$. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary quadrics the boundary of this locus is a degree forty hypersurface $J(σ_3(v_3(\mathbb{P}^3)),τ(…
We show that the degree of the images of the moduli space of (principally polarized) abelian varieties A_g and of the moduli space of curves M_g in the projective space under the theta constant embedding are equal to the top self-intersection numbers of one half the first Hodge class on them. This allows us to obtain an explicit formula for the degree of A_g, and an explicit upper bound for the degree of M_g. Knowing the degree of A_g allows us to effectively determine the subvariety itself, i.e. to effectively obtain all polynomial equations satisfied by theta constants. Furthermore, combinin…
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 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.
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…
Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formul…
Let $\mathcal{M}_{g,2}$ be the moduli space of curves of genus $g$ with a level-2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in $\mathcal{M}_{6,2}$. We prove also that for all $g\geqslant3$, each component of the hyperelliptic locus in $\mathcal{M}_{g,2}$ is a connected component of the intersection of $g-2$ thetanull divisors.
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 the codimension two locus H in A_g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class of H in A_g for every g. For g=4, this turns out to be the locus of Jacobians with a vanishing theta-null. For g=5, via the Prym map we show that H in A_5 has two components, both unirational, which we completely describe. This gives a geometric classification of 5-dimensional ppav whose theta-divisor has a quadratic singularity of non-maximal rank. We then determine the slope of the effective cone of …
Van Geemen and van der Geer, Donagi, Beauville and Debarre proposed characterizations of the locus of jacobians which use the linear system of $2Θ$-divisors. We give new evidence for these conjectures in the case of Prym varieties.
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the μ-ordinary locus, when applied to scalar-valued forms.
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…
Let SU(2,1) be the moduli space of stable rank two vector bundles having fixed determinant of odd degree over a compact Riemann surface C. In this paper it is shown that the Theta divisor of SU(2,1) is very ample for every C. The proof is related to the study of the base locus of the pencil of divisors 2-theta in the Jacobian of C which is naturally associated to a point of SU(2,1).
We compute the class of a divisor on M_{g,n} given as the closure of the locus of smooth pointed curves [C; x_1,..., x_n] for which \sum d_j x_j has an effective representative, where d_j are integers summing up to g-1, not all positive. The techniques used are a vector bundle computation, a pushdown argument reducing the number of marked points, and the method of test curves.
Mumford's well-known characterization of the hyperelliptic locus of the moduli space of ppavs in terms of vanishing and non-vanishing theta constants is based on Neumann's dynamical system. Poor's approach to the characterization uses the cross ratio. A key tool in both methods is Frobenius' theta formula, which follows from Riemann's theta formula. In a 2004 paper Grushevsky gives a different characterization in terms of cubic equations in second order theta functions. In this note we first show the connection between the methods by proving that Grushevsky's cubic equations are strictly relat…
We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.
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.
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
We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification \bar S_g of the moduli space S_g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of \bar S_g extends holomorphically to a desingularisation of \bar S_g.
We study the degree of the Gauss map of the theta divisor of principally polarised complex abelian varieties. We use this to obtain a bound on the multiplicity of the theta divisor along irreducible components of its singular locus, and apply this bound in examples, and to understand the local structure of isolated singular points. We further define a stratification of the moduli space of ppav's by the degree of the Gauss map. In dimension four, we show that this stratification gives a weak solution of the Schottky problem, and we conjecture that this is true in any dimension.
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 …
We carry out a complete birational classification of the universal theta divisor Th_g over the moduli space of curves of genus g, and show that Th_g enjoys good rationality properties for g<12, and is a variety of general type for g\geq 12. The key ingredient is an intersection-theoretic study of the universal antiramification locus of the Gauss map. We also present a complete classification of the universal symmetric product of degree g-2 over M_g.
We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the g…
To any closed subvariety $Y$ of a complex abelian variety one can attach a reductive algebraic group $G$ which is determined by the decomposition of the convolution powers of $Y$ via a certain Tannakian formalism. For a theta divisor $Y$ on a principally polarized abelian variety, this group $G$ provides a new invariant that naturally endows the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ with a finite constructible stratification. We determine $G$ for a generic principally polarized abelian variety, and for $g=4$ we show that the stratification detects the l…
We study moduli spaces of K3 surfaces endowed with a Nikulin involution and their image in the moduli space R_g of Prym curves of genus g. We observe a striking analogy with Mukai's well-known work on ordinary K3 surfaces. Many of Mukai's results have a very precise Prym-Nikulin analogue, for instance: (1) A general Prym curve from R_g is a section of a Nikulin surface if and only if g\leq 7, g\neq 6. (2) R_7 has the structure of a Mori fibre space over the corresponding moduli space of polarized Nikulin surfaces. (3) In the case of genus next to maximal (g=6), the Prym-Nikulin locus in R_6 is…
