Related papers: Normal Functions, Even Theta Characteristics and t…

We consider the partial theta function $\theta (q,z):=\sum _{j=0}^{\infty}q^{j(j+1)/2}z^j$, where $(q,z)\in \mathbb{C}^2$, $|q|<1$. We show that for any $0<\delta _0<\delta <1$, there exists $n_0\in \mathbb{N}$ such that for any $q$ with…

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…

Let $(\mathcal{L},\mathfrak{g})$ be a line bundle over a closed Riemann surface $(\Sigma,g)$, $\Gamma(\mathcal{L})$ be the set of all smooth sections, and $\mathcal{D}:\Gamma(\mathcal{L})\rightarrow T^\ast\Sigma\otimes \Gamma(\mathcal{L})$…

For the partial theta function $\theta (q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is…

Motivic and topological zeta functions are singularity invariants, mainly associated to a function $f$ and a top differential form $\omega$ on a smooth variety. When $\omega$ is the standard form $dx_1\wedge \dots \wedge dx_n$ on affine…

Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic…

We introduce an generalization of the theta divisor to the theory of holomorphic triples on a smooth projective curve $X$. We show that a given triple $T=(E_1 \to E_0)$ is $\alpha$-semistable iff there exists an orthogonal tripe $S=(F_1 \to…

Let $\{T(t)\}_{t\geq 0}$ be a $C_0$-semigroup of bounded linear operators on the Banach space ${X}$ into itself and let $A$ be their infinitesimal generator. In this paper, we show that if $T(t)$ is uniformly ergodic, then $A$ does not have…

We proved the factorization of generalized theta functions when the curve has two irreducible components meeting at one node.

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…

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

The Jacobian $J$ of a complete, smooth, connected curve $X$ admits a canonical divisor $\Theta$, called the Theta divisor. It is well-known that $\Theta$ is ample and, in fact, $3\Theta$ is very ample. For a general complete, integral curve…

In this paper we prove a strengthening of the generic vanishing result in characteristic $p>0$ given in [HP16]. As a consequence of this result, we show that irreducible $\Theta$ divisors are strongly F-regular and we prove a related result…

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

We classify finite dimensional $H_{m^2}(\zeta)$-simple $H_{m^2}(\zeta)$-module Lie algebras $L$ over an algebraically closed field of characteristic $0$ where $H_{m^2}(\zeta)$ is the $m$th Taft algebra. As an application, we show that…

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…

Let $C$ be a smooth projective curve of genus $g\geq 3$ and let $\eta$ be an odd theta characteristic on it such that $h^0(C,\eta) = 1$. Pick a point $p$ from the support of $\eta$ and consider the one-dimensional linear system $|\eta +…

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(\pi,G)_0$ denotes the null-component of the space of homomorphisms from $\pi$ to $G$, and $\mathrm{map}_*(B\pi,BG)_0$ denotes the null-component of…

Let the matching polynomial of a graph $G$ be denoted by $\mu (G,x)$. A graph $G$ is said to be $\theta$-super positive if $\mu(G,\theta)\neq 0$ and $\mu(G\setminus v,\theta)=0$ for all $v\in V(G)$. In particular, $G$ is 0-super positive if…