The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n.
A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every admissible slope is realized by a bundle which restricts to a semistable bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on P^1 which is interesting in its own right.
A Steiner bundle is a vector bundle on projective space arising as the cokernel of the map defined by a matrix of linear forms. These come up in various geometric settings, and by now they are the subject of a considerable literature. Starting with work of Dolgachev and Kapranov from 1993, several authors have considered the question of whether one can recover from the bundle the geometric data used to construct it. Here we prove such Torelli-type statements for the tautological bundles associated to sufficiently positive divisors on any very ample linear series. In an appendix, we give a new …
Let $f \colon (X,Δ) \to Y$ be a fibration such that $K_X + Δ$ is torsion along the fibres of $f$. Assume that $Y$ has dimension 2, or that $Y$ has dimension 3 and the fibres have dimension at most 3. Then the restriction of the moduli part to its augmented base locus is semiample.
In this article we deduce criteria for the splitting and the triviality of vector bundles, by restricting them to partially ample divisors. This allows to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with a number of examples. For products of minuscule homogeneous varieties, our results allow to test the splitting of vector bundles by restricting them to products of Schubert 2-planes. The triviality criteria obtained inhere are particularly suited to Frobenius split varieties, whose splitting is defined by a section in the anti-canonical li…
For a semistable reflexive sheaf $E$ of rank $r$ and $c_1=a$ on $¶^n$ and an integer $d$ such that $r|ad$, we give sufficient conditions so that the restriction of $E$ on a generic rational curve of degree $d$ is balanced, i.e. a twist of the trivial bundle (for instance, if $E$ has balanced restriction on a generic line, or $r=2$ or $E$ is an exterior power of the tangent bundle). Assuming this, we give a formula for the 'virtual degree', interpreted enumeratively, of the locus of rational curves of degree $d$ on which the restriction of $E$ is not balanced, generalizing a classical formula d…
Inside the projectivized $k$-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence geometry and restrictions to various families, including pencils of curves on K3 surfaces and pencils of Du Val curves. We also show the extremality and rigidity of the closure of the incidence divisor consisting of smooth pointed curves together with a canonical or 2-canonical divisor passing through the marked point.
We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if t…
For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.
The restriction, on the spectral variables, of the Baker-Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of differential operators in $k$ variables. The remaining g-k derivations define evolution equations for generators of the BA-module. As a corollary new examples of commutative ring of partial differential operators with matrix coefficients and their non-trivial evolution equ…
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $Ø_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $Δ(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.
In this paper for each $n\ge g\ge 0$ we consider the moduli stack $\widetilde{\mathcal U}^{ns}_{g,n}$ of curves $(C,p_1,\ldots,p_n,v_1,\ldots,v_n)$ of arithmetic genus $g$ with $n$ smooth marked points $p_i$ and nonzero tangent vectors $v_i$ at them, such that the divisor $p_1+\ldots+p_n$ is nonspecial (has no $h^1$) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian $G(n-g,n)$. We also construct an isomorphism of $\widetilde{\mathcal U}^{ns}_{g,n}$ with a certain relative moduli of $A_\infty$-structures (up to an equivalence)…
We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture of Peternell. As a consequence, the second Chern class of such a manifold has non-negative intersections with ample divisors. We also investigate under which conditions these positivities are strict, and answer a question of Yau.
We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.
We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an isomorphism for a general member $Y\in |L|$ provided that $\dim{X}\geq 4$. This is a generalization of the Grothendieck-Lefschetz Theorem, for divisor class groups of singular varieties.
Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n\geq 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $|\mathcal{L}|$ contains an element $A=A_{1}+...+A_{r}, r\geq 1$, where all the $A_{i}$'s are distinct effective Cartier divisors with Pic$(A_i)=\mathbb{Z}$, we show that such a $\mathcal{V}$ is as special as the components $A_i$ of $A\in |\mathcal{L}|$. After making a list of some consequences about the positivity of the components $A_i$, we characterize pairs $(\mathcal{V}, \mathcal{L})$ as above when e…
In this paper we study holomorphic rank two vector bundles on the blow up of $ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck says that any holomorphic vector bundle on the projective plane ${\bf P}^1$ splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits. Moreover we assume that $E$ is a rank two bundle that has zero first Chern class. Hence its restriction to the exceptional divisor is of…
We prove that, for a good meromorphic flat bundle with poles along a divisor with normal crossings, the restriction of the irregularity complex to each natural stratum of this divisor only depends on the formal flat bundle along this stratum. This answers a question raised by J.-B. Teyssier. This version takes into account the published erratum.
We study restriction of logarithmic Higgs bundles to the boundary divisor and we construct the corresponding nearby-cycles functor in positive characteristic. As applications we prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures and their generalizations. This implies, e.g., semipositivity for the relative canonical divisor of a semistable reduction in positive characteristic and it gives some new strong results generalizing semipositivity even for complex varieties.
In the paper \cite{Lau16}, it was shown that the restriction of a pseudoeffective divisor $D$ to a subvariety $Y$ with nef normal bundle is pseudoeffective. Assuming the normal bundle is ample and that $D|_Y$ is not big, we prove that the numerical dimension of $D$ is bounded above by that of its restriction, i.e. $κ_σ(D)\leq κ_σ(D|_Y)$. The main motivation is to study the cycle classes of "positive" curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movab…
With every smooth, projective algebraic curve $\tilde{C}$ with involution $σ:\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-σ)J(\tilde{C})$ with principal polarization $Ξ$ such that $2Ξ$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $Θ$ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},σ)$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We con…
For $n\geq 3$, let $M$ be an $(n+r)$-dimensional irreducible Hermitian symmetric space of compact type and let $\mathcal{O}_M(1)$ be the ample generator of $Pic(M)$. Let $Y=H_1\cap\dots\cap H_r$ be a smooth complete intersection of dimension $n$ where $H_i\in\vert \mathcal{O}_M(d_i)\vert$ with $d_i\geq 2$. We prove a vanishing theorem for twisted holomorphic forms on $Y$. As an application, we show that the tangent bundle $T_Y$ of $Y$ is stable. Moreover, if $X$ is a smooth hypersurface of degree $d$ in $Y$ such that the restriction $Pic(Y)\rightarrow Pic(X)$ is surjective, we establish some e…
Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument g…
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any principal divisor on a tropical curve is the restriction of a principal divisor on the ambient plane $\mathbb{R}^2$.
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
Consider the blow-up $X$ of $\mathbb{P}^3$ at 6 points in very general position and the 15 lines through the 6 points. We construct an infinite-order pseudo-automorphism $ϕ_X$ on $X$, induced by the complete linear system of a divisor of degree 13. The effective cone of $X$ has infinitely many extremal rays and hence, $X$ is not a Mori Dream Space. The threefold $X$ has a unique anticanonical section which is a Jacobian K3 Kummer surface $S$ of Picard number 17. The restriction of $ϕ_X$ on $S$ realizes one of Keum's 192 infinite-order automorphisms of Jacobian K3 Kummer surfaces. In general, w…
We show that there exists a surjection from the set of effective divisors of degree $g$ on a tropical curve of genus $g$ to its Jacobian by using a tropical version of the Riemann-Roch theorem. We then show that the restriction of the surjection is reduced to the bijection on an appropriate subset of the set of effective divisors of degree $g$ on the curve. Thus the subset of effective divisors has the additive group structure induced from the Jacobian. We finally realize the addition in Jacobian of a tropical hyperelliptic curve of genus $g$ via the intersection with a tropical curve of degre…
The purpose of this paper is to explore the geometry and establish the slope stability of tautological vector bundles on Hilbert schemes of points on smooth surfaces. By establishing stability in general we complete a series of results of Schlickewei and Wandel who proved the slope stability of these vector bundles for Hilbert schemes of 2 points or 3 points on K3 or abelian surfaces with Picard group restrictions. In exploring the geometry we show that every sufficiently positive semistable vector bundle on a smooth curve arises as the restriction of a tautological vector bundle on the Hilber…
Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.
Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction $\mathcal{U}_{L,x}$ of the Poincare bundle to $x \times U_Y(n,d,L)$ is stable with respect to the polarisation $θ_L$ and its restriction to $x \times U'_Y(n,d,L)$, where $U'_Y(n,d,L)$ is the moduli space of vector bund…
Inspired by the log Gromov-Witten (or GW) theory of Gross-Siebert/Abramovich-Chen, we introduce a geometric notion of log J-holomorphic curve relative to a simple normal crossings symplectic divisor defined in [FMZ1]. Every such moduli space is characterized by a second homology class, genus, and contact data. For certain almost complex structures, we show that the moduli space of stable log J-holomorphic curves of any fixed type is compact and metrizable with respect to an enhancement of the Gromov topology. In the case of smooth symplectic divisors, our compactification is often smaller than…
Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let Λbe its value group. Given a smooth, proper, connected K-curve X and a skeleton Γof the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(Γ) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classic…
In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrictions of line bundles on the variety. As a consequence, we describe the group of sections of the Hitchin fibration for moduli spaces of Higgs bundles on curves.
We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of ``movable curves'', which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive (1,1)-currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is is not uniruled. We also prove that a 4-fold with a canonical …
Let Sigma be a smooth complex curve, and let S be the product ruled surface Sigma \times CP^1. We prove a correspondence conjectured by Donaldson between finite energy U(2)-instantons over the cylinder Sigma \times S^1 \times R, and rank 2 holomorphic bundles over S whose restrictions to the divisors at infinity are stable.
Let $X$ be an irreducible smooth projective curve of genus $g\ge3$ defined over the complex numbers and let ${\mathcal M}_ξ$ denote the moduli space of stable vector bundles on $X$ of rank $n$ and determinant $ξ$, where $ξ$ is a fixed line bundle of degree $d$. If $n$ and $d$ have a common divisor, there is no universal vector bundle on $X\times {\mathcal M}_ξ$. We prove that there is a projective bundle on $X\times {\mathcal M}_ξ$ with the property that its restriction to $X\times\{E\}$ is isomorphic to $P(E)$ for all $E\in\mathcal{M}_ξ$ and that this bundle (called the projective Poincaré bu…
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${\mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known that ${\mathcal N}_H(r,d)$ has a natural holomorphic Poisson structure which is in fact symplectic if and only if $D$ is the zero divisor. We prove that ${\mathcal N}_H(r,d)$ admits a natural enhancement to a holomorphic symplectic manifold which is called here ${\mathcal M}_H(r,d)$. This ${\mathcal M}_H(r,d)$ is constructed by trivializing, over…
Topologically, a compact Riemann surface $X$ of genus $g$ is a $g$-holed torus (a sphere with $g$ handles). This paper is an introduction to the theory of compact Riemann surfaces and algebraic curves. It presents the basic ideas and properties as an expository essay, explores some of their numerous consequences and gives a concise account of the elementary aspects of different viewpoints in curve theory. We discuss and prove most intuitively some geometric-topological aspects of the algebraic functions and the associated Riemann surfaces. Abelian and normalized differentials, Riemann's biline…
We describe a relation between the invariants of $n$ ordered points in $P^d$ and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2} \subset P^d$. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cove…
