In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.
The purpose of this paper is to study Seshadri constants on the self-product $E\times E$ of an elliptic curve $E$. We provide explicit formulas for computing the Seshadri constants of all ample line bundles on the surfaces considered. As an application, we obtain a good picture of the behaviour of the Seshadri function on the nef cone.
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 locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. We also study the case of arbitrary codimension.
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 Seshadri constants and geometry of surfaces.
In this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $E_1\times E_2$ without complex multiplication are integers. By studying elliptic curves on $E_1\times E_2$ we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $E_1\times E_2$ are integers if and only if the minimal degree of an isogeny $E_1\to E_2$ equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $E_1\times E_2$.
Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the rank or semistability of $E_1$ and $E_2$. We also study the Seshadri constants of ample line bundles on $X$. We calculate the Seshadri constants in some cases and give bounds in some of the remaining cases.
Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$, let $\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of Seshadri constant. Our results generalize various known results for ${\rm rank}(E)=2$.
So far, Seshadri constants on abelian surfaces are completely understood only in the cases of Picard number one and on principally polarized abelian surfaces with real multiplication. Beyond that, there are partial results for products of elliptic curves. In this paper, we show how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. We develop an effective algorithm depending only on the basis of the Néron-Severi group to compute not only the Seshadri constants but also the numerical data of their Seshadri curves. Access to the Seshadri curv…
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 result of Ciliberto-Kouvidakis that reduces finding the ample cone of X_g to the Nagata conjecture when g\ge 9. We also give new bounds on the the ample cone of X_g when g=5.
We compute in this note the Seshadri constants of the anticanonical bundle at every point of Del Pezzo surfaces. During the proof, we enlight the role of rational curves in our computations. We present then two exemples where the positivity of the anticanonical bundle cannot be detected using rational curves.
In "Seshadri fibrations of algebraic surfaces" [arXiv:0709.2592v1] we showed that if the multiple point Seshadri constants of an ample line bundle on a smooth projective surface in very general points satisfy certain inequality then the surface is fibred by curves computing these constants. Here we characterize the border case of polarized surfaces whose Seshadri constants in general points fulfill the equality instead of inequality and which are not fibred by Seshadri curves. It turns out that these surfaces are the projective plane and surfaces of minimal degree.
In this paper we consider the question of when Seshadri constants on abelian surfaces are integers. Our first result concerns self-products $E\times E$ of elliptic curves: If $E$ has complex multiplication in $\Z[i]$ or in $\Z[\frac12(1+i\sqrt3)]$ or if $E$ has no complex multiplication at all, then it is known that for every ample line bundle $L$ on $E\times E$, the Seshadri constant $\eps(L)$ is an integer. We show that, contrary to what one might expect, these are in fact the only elliptic curves for which this integrality statement holds. Our second result answers the question how -- on an…
We examine how the Seshadri constant of an ample line bundle at a very general point of an algebraic surface can carry important global geometric information about the surface. In particular, we obtain a numerical criterion for when a surface admits a dominant map to an algebraic curve.
We define the Seshadri constant of a space curve and consider ways to estimate it. We then show that it governs the gonality of the curve. We use an argument based on Bogomolov's instability theorem on a threefold. The same methods are then applied to the study of the behaviour of a stable vector bundle on P^3 under restriction to curves and surfaces.
We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose ta…
One of Demailly's characterizations of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note sections of multiples of the line bundle are used to produce such metrics and then to deduce another formula for Seshadri constants. It is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture.
We prove that the Seshadri constant of a polarized abelian variety is equal to the Seshadri constant of its abelian subvariety if the Seshadri constant is relatively small with respect to its degree, or it contains an abelian divisor which has sufficiently small degree. As an application of these results, we show that the Seshadri constant of a polarized abelian surface is computed by the unique Seshadri curve if the Seshadri constant is small enough. As another application, we compute the Seshadri constants of some non-simple polarized abelian threefolds which contain principally polarized ab…
Chennai Mathematical Institute（钦奈数学研究所）；Indian Institute of Technology (Indian School of Mines) Dhanbad（印度技术学院（印度矿业学院）丹巴德分校）；Indraprastha Institute of Information Technology, Delhi（德里因德拉普拉斯塔信息技术学院）
Let $X$ be a smooth projective variety defined over a field $k$ of characteristic $0$ and let $\mathcal{L}$ be a nef line bundle defined over $k$. We prove that if $x\in X$ is a $k$-rational point then the Seshadri constant $ε(X, \mathcal{L}, x)$ over $\overline{k}$ is the same as that over $k$. We show, by constructing families of examples, that there are varieties whose global Seshadri constant $ε(X)$ is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.
We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.
We study the Seshadri constants of cyclic coverings of smooth surfaces. The existence of an automorphism on these surfaces can be used to produce Seshadri exceptional curves. We give a bound for multiple Seshadri constants on cyclic coverings of surfaces with Picard number 1. Morevoer, we apply this method to $n$-cyclic coverings of the projective plane. When $2\leq n\leq 9$, explicit values are obtained. We relate this problem with the Nagata conjecture.
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
We prove two new results for Seshadri constants on surfaces of general type. Let $X$ be a surface of general type. In the first part, inspired by \cite{B-S}, we list the possible values for the multi-point Seshadri constant $\varepsilon(K_X,x_1,x_2,...,x_r)$ when it lies between $0$ and $1/r$, where $K_X$ is the canonical line bundle on $X$. In the second part, we assume $X$ of the form $C \times C$, where $C$ is a general smooth curve of genus $g \geq 2$. Given such $X$ and an ample line bundle $L$ on $X$ with some conditions on it, we show that the global Seshadri constant of $L$ is a ration…
Let $Y$ be a submanifold of dimension $y$ of a polarized complex manifold $(X,A)$ of dimension $k\geq 3$, with $1\leq y\leq k-1$. We define and study two positivity conditions on $Y$ in $(X,A)$, called Seshadri $A$-bigness and (a stronger one) Seshadri $A$-ampleness. In this way we get the natural generalization of the theory initiated by Paoletti in \cite{Pao} (which corresponds to the case $(k,y)=(3,1)$) and subsequently generalized and completed in \cite{BBF} (regarding curves in a polarized manifold of arbitrary dimension). The theory presented here, which is new even if $y=k-1$, is motiva…
In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.
Motivated by a similar result of Dumnicki, Küronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface $X$ obtained by blowing up very general points of $\mathbb{P}^2_\mathbb{C}$, assuming only that all prime divisors on $X$ of negative self-intersection are smooth rational curves $C$ with $C^2=-1$. (This assumption is a consequence of the SHGH Conjecture, but it is weaker than assuming the full conjecture.)
In the present note, we focus on certain properties of special curves that might be used in the theory of multi-point Seshadri constants for ample line bundles on the complex projective plane. In particular, we provide three Ein-Lazarsfeld-Xu-type lemmas for plane curves and a lower bound on the multi-point Seshadri constant of $\mathcal{O}_{\mathbb{P}^{2}}(1)$ under the assumption that the chosen points are not very general. In the second part, we focus on certain arrangements of points in the plane which are given by line arrangements. We show that in some cases the multi-point Seshadri cons…
We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.
Let $X$ be a smooth projective curve of genus $g\ge 3$ and $M_0$ be the moduli space of rank 2 semistable bundles over $X$ with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan \cite{NR}, Seshadri \cite{Se1} and Kirwan \cite{k5} respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwan's desingularization to Seshadri's, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of $M_0$ …
The purpose of this paper is to explicitly compute the Seshadri constants of all ample line bundles on fake projective planes. The proof relies on the theory of the Toledo invariant, and more precisely on its characterization of $\C$-Fuchsian curves in complex hyperbolic spaces.
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 divisors. The link is established via symplectic blowing up, in the spirit of McDuff and Polterovich. As an application of the main inequality, we get a simple new proof of a statement of Buser-Sarnak type to the effect that the Jacobian of a curve has a period of unusually small length. We also deduce a lower bound on the Seshadri constant of a very general p.…
Let $\mathbb K$ be an algebraically closed field of characteristic $0$. A curve of $(\mathbb K^*)^2$ arising from a Laurent polynomial in two variables is {\em intrinsic negative} if its tropical compactification has negative self-intersection. The aim of this note is to start a systematic study of these curves and to relate them with the problem of computing Seshadri constants of toric surfaces.
Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the number of components of ${\rm Curves}_d(X,x)$ for any smooth point $x$ outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on $n$ and $d$. An effective bound is given. A key ingredient of the proof is an argument from Ein-Küchle-Lazarsfeld's work on Seshadri numbers.
Using Dumnicki's approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on $\mathbb{P}^2$ we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on $\mathbb{P}^2$. With this method we prove the lower bound 4/13 for 10 general points on $\mathbb{P}^2$.
Over a family $\mathbb X$ of genus $g$ complete curves, which gives the degeneration of a smooth curve into one with nodal singularities, we build a moduli space which is the moduli space of ${\rm SL}(n, \mathbb C)$ bundles over the generic smooth curve $X_t$ in the family, and is a moduli space of bundles equipped with extra structure at the nodes for the nodal curves in the family. This moduli space is a quotient by $(\mathbb C^*)^s$ of a moduli space on the desingularisation. Taking a "maximal" degeneration of the curve into a nodal curve built from the glueing of three-pointed spheres, we …
We realize any space of conformal blocks attached to a punctured curve inside the cohomology of a configuration space of that curve and compare the WZW connection with the Gauss-Manin connection.
We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the Fano surfaces according to the automorphism group generated by these involutions and determine the configurations of their genus 2 curves. We study the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves generate a rank 25=h^{1,1} index 2 subgroup of the Néron-Severi group.
We identify certain Gromov-Witten invariants counting rational curves with given incidence and tangency conditions with the Betti numbers of moduli spaces of point configurations in projective spaces. On the Gromov-Witten side, S. Fomin and G. Mikhalkin established a recurrence relation via tropicalization, which is realized on the moduli space side using Donaldson-Thomas invariants of subspace quivers.
In this paper we consider plane quartics with to involutions. We compute the Dixmier invariants, the bitangents and the Matrix representation problem of these curves, showing that they have symbolic solutions for the last two questions.
The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.
We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$.
Similarly to the linear Harbourne constant recently defined, we study the elliptic $H$-constants of $\mathbb{P}^{2}$ and Abelian surfaces. We exhibit configurations of smooth plane cubic curves whose Harbourne index is arbitrarily close to $-4$. As a Corollary, we obtain that the global $H$-constant of any surface $X$ is less or equal to $-4$. Related to these problems, we moreover give a new inequality for the number and multiplicities of singularities of elliptic curves arrangements on Abelian surfaces, inequality which has a close similarity to the one of Hirzebruch for arrangements of line…
We give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections $X \subset \mathbb{P}^{n+2}_{\mathbb{C}}$ of multi-degree $d=(d_1, \dots, d_n)$, and we provide a sharp and uniform bound on their H-constants, which only depends on $d$.
A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $…
