arXiv subjects
Publications and source records attributed to Riccardo Salvati Manni.
We study theta series with half-integral characteristics attached to odd positive-definite unimodular lattices. In even rank, we get non-vanishing theta series with odd characteristics. We emphasize ranks divisible by four, where the involved rational lattices are integral unimodular neighbors.
We determine the maximal dimension of compact subvarieties of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties of dimension $g$, and the maximal dimension of a compact subvariety through a very general point of $\mathcal{A}_g$. This also allows us to draw some conclusions for compact subvarieties of the moduli space of complex curves of compact type.
We study the Kodaira dimension of the compactified n-fold Kuga variety over the moduli space of principally polarised abelian g-folds. We construct a suitable compactification, which we call a Namikawa compactification, and show that in most cases it has canonical singularities. We then use results about the slope of Siegel modular forms to determine the Kodaira dimension for all g>1 and n>0.
We consider the space of Siegel modular forms of genus $g$ of weight two relative to the main congruence subgroup of level 2 and to Igusa's group $Γ_g(4, 8)$ and $Γ_g(2,4)$.One of the main results of this paper is that in the case $g\ge 8$ the space $[Γ_g[4,8],2]$ is generated by the products of 4 theta nullwerte. Thus this note can be considered as a completion of the example at the end of [Fr].
In [ Ge], Bert van Geemen computed the dimension of the space of the fourth power of the theta nullwerte. In [SM2], it has been observe that all linear relations between the $θ_m^4$ are consequences of the quartic Riemann relations. In this note, we want to give a new proof of these result and extend them. In a last section we treat the linear dependencies between arbitrary powers $\vartheta[m]^k$. We will show that $k=4$ is the only case where such dependencies can occur. For this reason, we give a slightly different title: Some remarks to a Theorem of van Geemen
A subvariety of a quasi-projective complex variety $X$ is called ``universally irreducible'' if its preimage inside the universal cover of $X$ is irreducible. In this paper we investigate sufficient conditions for universal irreducibility. We consider in detail complete intersection subvarieties of small codimension inside Siegel moduli spaces of any finite level. Moreover we show that, for $g\geq 3$, every Siegel modular form is the product of finitely many irreducible analytic functions on the Siegel upper half-space $\mathbb{H}_g$. We also discuss the special case of singular theta series of weight $\frac{1}{2}$ and of Schottky forms.
We study various naturally defined subvarieties of the moduli space ${\mathcal A}_g$ of complex principally polarized abelian varieties (ppav) in a neighborhood of the locus of products of $g$ elliptic curves. In this neighborhood, we obtain a local description for the locus of hyperelliptic curves, reproving the recent result of Shepherd-Barron that the hyperelliptic locus is locally given by tridiagonal matrices. We further reprove and generalize to arbitrary genus the recent result of Agostini and Chua showing that the locus of Jacobians of genus 5 curves with a theta-null is an irreducible component of the locus of ppav with a theta-null such that the singular locus of the theta divisor at the corresponding two-torsion point has tangent cone of rank at most 3. We further show that the locus of ppav such that the gradient vanishes, for some odd theta characteristic, locally has codimension $g$ near the diagonal. Finally, we obtain new results on the locus where the rank of the Hessian of the theta function at a two-torsion point that lies on the theta divisor is equal to 2.
We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on ${\mathcal A}_g$ for $g\leq 4$, and gives an explicit upper bound for the moving slope of ${\mathcal A}_5$ and a conjectural upper bound for the moving slope of ${\mathcal A}_6$.
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 A_5 and show that the component N_0' of the Andreotti-Mayer divisor has minimal slope 54/7. Furthermore, the Iitaka dimension of the linear system corresponding to N_0' is submaximal.
We construct a basis of the space ${\text S}_{14}({\text{Sp}}_{12}({\mathbb Z}))$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic theta series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of $\mathcal{A}_6$ is non-negative.
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.
In this note we prove a sharp bound for the number of 2-torsion points on a theta divisor and show that this is achieved only in the case of products of elliptic curves. This settles in the affirmative a conjecture of Marcucci and Pirola.
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 related to Frobenius' theta formula and we then give a new proof of Mumford's characterization via Gunning's multisecant formula.
Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well.
Using the irreducibility of a natural irreducible representation of the theta group of an ample line bundle on an abelian variety, we derive a bound for the number of $n$-torsion points that lie on a given theta divisor. We present also two alternate approaches to attacking the case $n=2$.
In this paper we will prove that the six-dimensional Göpel variety in $P^{134}$ is generated by 120 linear, 35 cubic and 35 quartic relations. This result was already obtained in [RS] , but the authors used a statement in [Co] saying that the Göpel variety set theoretically is generated by the linear and cubic relations alone. Unfortunately this statement is false. There are 120 extra points. Nevertheless the results stated in [RS] are correct. There are required several changes that we will illustrate in some detail
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 image. This is equivalent to the statement that the image is a normal subvariety of $P^{35}$ .
We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.