This book can form the basis of a second course in algebraic geometry. As motivation, it takes concrete questions from enumerative geometry and intersection theory, and provides intuition and technique, so that the student develops the ability to solve geometric problems. The authors explain key ideas, including rational equivalence, Chow rings, Schubert calculus and Chern classes, and readers will appreciate the abundant examples, many provided as exercises with solutions available online. Intersection is concerned with the enumeration of solutions of systems of polynomial equations in several variables. It has been an active area of mathematics since the work of Leibniz. Chasles' nineteenth-century calculation that there are 3264 smooth conic plane curves tangent to five given general conics was an important landmark, and was the inspiration behind the title of this book. Such computations were motivation for Poincare's development of topology, and for many subsequent theories, so that intersection theory is now a central topic of modern mathematics.

3264 and All That

Basic Algebraic Geometry

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

Lectures on Hilbert schemes of points on surfaces

Cambridge University Press v. Georgia State University

We use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: 
1. We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. 
2. We establish a long-standing conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the Beauville-Bogomolov form). 
These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wall-crossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgeland-stable objects on X.

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations

The minimal model program for the Hilbert scheme of points on P2 and Bridgeland stability

This paper studies the geometry of one-parameter specializations of subvarieties of Grassmannians and two-step flag varieties. As a consequence, we obtain a positive, geometric rule for expressing the structure constants of the cohomology of two-step flag varieties in terms of their Schubert basis. A corollary is a positive, geometric rule for computing the structure constants of the small quantum cohomology of Grassmannians. We also obtain a positive, geometric rule for computing the classes of subvarieties of Grassmannians that arise as the projection of the intersection of two Schubert varieties in a partial flag variety. These rules have numerous applications to geometry, representation theory and the theory of symmetric functions.

/pdf/a-littlewood-richardson-rule-for-two-step-flag-varieties-1oi4dk4zo9.pdf

A Littlewood–Richardson rule for two-step flag varieties

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in of degree 2d ≤ min (n + 4, 2n − 2) and dimension at least three is irreducible and of the expected dimension.

/pdf/rational-curves-on-smooth-cubic-hypersurfaces-1bteoh3b3j.pdf

Rational Curves on Smooth Cubic Hypersurfaces

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.

https://www.springer.com/cda/content/document/cda_downloaddocument/9781461464815-c1.pdf?SGWID=0-0-45-1445120-p174843536

The Birational Geometry of the Hilbert Scheme of Points on Surfaces

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.

/pdf/degenerations-of-surface-scrolls-and-the-gromov-witten-23wq166f4l.pdf

Izzet Coskun

Papers

The minimal model program for the Hilbert scheme of points on P2 and Bridgeland stability

A Littlewood–Richardson rule for two-step flag varieties

Rational Curves on Smooth Cubic Hypersurfaces

The Birational Geometry of the Hilbert Scheme of Points on Surfaces

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