Related papers: Remarks on Seshadri constants

We study families of curves covering a projective surface and give lower bounds on the self-intersection of the members of such families, improving results of Ein-Lazarsfeld and Xu. We apply the obtained inequalities to get new insights on…

We study the hyperbolicity of the log variety $(\mathbb{P}^n, X)$, where $X$ is a very general hypersurface of degree $d\geq 2n+1$ (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of…

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the…

Let $X$ be the moduli space of semistable rank 2 vector bundles over a smooth curve C of genus $g \ge 2$ and $\theta : X \to PH^0(L)^*$ be the map associated to the generalized theta divisor L on X. We prove that for C not hyperelliptic,…

Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $\Delta=S+2\,T.$ We assume $\deg\Delta=6.$ Let $\pi:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $\Delta,$ i.e. $\pi$ is ramified…

Let $E$ be a holomorphic vector bundle on a compact K\"ahler manifold $X$. If we fix a metric $h$ on $E$, we get a Laplace operator $\Delta$ acting upon smooth sections of $E$ over $X$. Using the zeta function of $\Delta$, one defines its…

Given a smooth projective curve X, we give effective very ampleness bounds for generalized theta divisors on the moduli spaces $SU_X(r,d)$ and $U_X(r,d)$ of semistable vector bundles of rank r and degree d on X with fixed, respectively…

We discuss stability of square root domains for uniformly elliptic partial differential operators $L_{a,\Omega,\Gamma} = -\nabla\cdot a \nabla$ in $L^2(\Omega)$, with mixed boundary conditions on $\partial \Omega$, with respect to additive…

Given a closed subscheme $Z$ of a polarized abelian variety $(A,\ell)$ we define its vanishing threshold with respect to $\ell$ and relate it to the Seshadri constant of the ideal defining $Z.$ As a particular case, we introduce the notion…

We prove that projectivity is an open condition for deformations of algebraic spaces with rational singularities. V.2: references updated and corrected.

We give an asymptotic formula for the number of $\mathbb{F}_{q}$-rational points over a fixed determinant moduli space of stable vector bundles of rank $r$ and degree $d$ over a smooth, projective curve $X$ of genus $g \geq 2$ defined over…

Let X_g=C^{(2)}_g be the second symmetric product of a very general curve of genus g. We reduce the problem of describing the ample cone on X_g to a problem involving the Seshadri constant of a point on X_{g-1}. Using this we recover a…

We prove that: I. For every regular Lindel\"of space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|\ne\omega$, then $X$ is maximally resolvable; II. For every regular countably compact space $X$ if $|X|=\Delta(X)$ and $\mathrm{cf}|X|=\omega$,…

To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the…

We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base…

We establish an improvement of Philippon's zero estimates primarily in the multiplicity setting. The improvement is made possible by a more geometric approach and in particular the use of Seshadri constants.

The purpose of this note is to point out an elementary but somewhat surprising connection between the work of Buser and Sarnak on lengths of periods of abelian varieties and the Seshadri constants measuring the local positivity of theta…

Let $X$ be a normal projective variety equipped with an action of a semisimple algebraic group $G$, and assume that $X$ contains a unique closed orbit. Let $B$ be a Borel subgroup of $G$ and let $E$ be a $B$-equivariant vector bundle on…

We prove that classes of rational curves on very general Enriques surfaces are always $2$-divisible. As a consequence, we prove that the Seshadri constant of any big and nef line bundle on a very general Enriques surface coincides with the…

We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a…