Izzet Coskun

Degenerations of surface scrolls and the Gromov-Witten invariants of Grassmannians

We describe an algorithm for computing certain characteristic numbers of rational normal surface scrolls using degenerations. As a corollary we obtain an efficient method for computing the corresponding Gromov-Witten invariants of the Grassmannians of lines.

The Birational Geometry of the Hilbert Scheme of Points on Surfaces

In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as \({\mathbb{P}}^{2}, {\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) and \(\mathbb{F}_{1}\). We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K 2≥2. As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland-stable objects. When the surface is \({\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) or \(\mathbb{F}_{1}\), we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of Arcara et al. (The birational geometry of the Hilbert scheme of points on \({\mathbb{P}}^{2}\) and Bridgeland stability, arxiv:1203.0316, 2012) to these surfaces.

The Effective Cone of the Kontsevich Moduli Space

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, M0,0(P r , d), stabilize when r ≥ d. We give a complete characterization of the effective divisors on M0,0(P d , d). They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607 e-mail: coskun@math.uic.edu Department of Mathematics, Stony Brook University, Stony Brook, NY 11794 e-mail: jstarr@math.sunysb.edu Department of Mathematics, Harvard University, Cambridge, MA 02138 e-mail: harris@math.harvard.edu Received by the editors May 26, 2006. The second author was partially supported by the NSF grant DMS-0200659. The third author was partially supported by the NSF grant DMS-0353692 and a Sloan Research Fellowship. AMS subject classification: Primary: 14D20; secondary: 14E99, 14H10. c ©Canadian Mathematical Society 2008. 519

The effective cone of the moduli space of sheaves on the plane

We compute the cone of effective divisors on any moduli space of semistable sheaves on the plane. The computation hinges on finding a good resolution of a general stable sheaf. This resolution is determined by Bridgeland stability and arises from a well-chosen Beilinson spectral sequence. The existence of a good choice of spectral sequence depends on remarkable number-theoretic properties of the slopes of exceptional bundles.

Extremal higher codimension cycles on moduli spaces of curves

We show that certain geometrically defined higher codimension cycles are ex- tremal in the effective cone of the moduli space Mg,n of stable genus g curves with n ordered marked points. In particular, we prove that codimension two boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in M3,1 and the locus of hyperel- liptic curves in M4 are extremal cycles. In addition, we exhibit infinitely many extremal codimension two cycles in M1,n for n � 5 and in M2,n for n � 2.

The Ample Cone of the Kontsevich Moduli Space

We produce ample (resp. NEF, eventually free) divisors in the Kontsevich space M0,n(P r , d) of n-pointed, genus 0, stable maps to P r , given such divisors in M0,n+d. We prove that this produces all ample (resp. NEF, eventually free) divisors in M0,n(Pr, d). As a consequence, we con- struct a contraction of the boundary S⌊d/2⌋ k=1 �k,d−k in M0,0(Pr, d), analogous to a contraction of the

Towards Mori's program for the moduli space of stable maps

We introduce and compute the class of a number of effective divisors on the moduli space of stable maps

The ample cone of moduli spaces of sheaves on the plane

\overline{{\scr M}}_{0,0}({\Bbb P}^{r}, d)

The nef cone of the moduli space of sheaves and strong Bogomolov inequalities

, which, for small

Divisors on the space of maps to Grassmannians

d