Three questions in Gromov-Witten theory

This article accompanies my ICM talk in August 2002. Three conjectural directions in Gromov-Witten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood.Comment: 10 page

The Chow Ring of the Hilbert Scheme of Rational Normal Curves

Let H(d) be the (open) Hilbert scheme of rational normal curves of degree d in P^d. A presentation of the integral Chow ring of H(d) is given via equivariant Chow ring computations. Included also in the paper are algebraic computations of the integral equivariant Chow rings of the algebraic groups O(n), SO(2k+1). The results for S0(3)=PGL(2) are needed for the Hilbert scheme calculation.Comment: 24 pages, Latex2

Rational curves on hypersurfaces [after A. Givental]

This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating hypergeometric series to solutions of quantum differential equations arising from hypersurfaces in projective space. A particular case of this relationship is a proof of the Mirror prediction for the numbers of rational curves on the Calabi-Yau quintic 3-fold. The approach taken here is entirely algebro-geometric and relies upon a localization formula on the moduli space of stable genus 0 maps to projective space. A different proof of the quintic Mirror prediction may be found in the work of Lian, Liu, and Yau.Comment: 33 pages, Latex2

Convex rationally connected varieties

Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained by an elementary argument.Comment: 7 page

A Note On Elliptic Plane Curves With Fixed j-Invariant

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to be {d-1 \choose 2}*N_d.Comment: 10 pages, AMSLate

A calculus for the moduli space of curves

This article accompanies my lecture at the 2015 AMS summer institute in algebraic geometry in Salt Lake City. I survey the recent advances in the study of tautological classes on the moduli spaces of curves. After discussing the Faber-Zagier relations on the moduli spaces of nonsingular curves and the kappa rings of the moduli spaces of curves of compact type, I present Pixton's proposal for a complete calculus of tautological classes on the moduli spaces of stable curves. Several open questions are discussed. An effort has been made to condense a great deal of mathematics into as few pages as possible with the hope that the reader will follow through to the end.Comment: 39 pages, 7 diagram

The kappa classes on the moduli spaces of curves

In the past few years, substantial progress has been made in the understanding of the algebra of kappa classes on the moduli spaces of curves. My goal here is to provide a short introduction to the new results. Along the way, I will discuss several open questions. The article accompanies my talk at "A celebration of algebraic geometry" at Harvard in honor of the 60th birthday of J. Harris.Comment: 10 page

A Geometric Invariant Theory Compactification of M_{g,n} Via the Fulton-MacPherson Configuration Space

A compactification over $\overline{M}_g$ of $M_{g,n}$ is obtained by considering the relative Fulton-MacPherson configuration space of the universal curve. The resulting compactification differs from the Deligne-Mumford space $\overline{M}_{g,n}$. In case $n=2$, the compactification constructed here and the Deligne-Mumford compactification are essentially the distinct minimal resolutions of the fiber product over $\overline{M}_g$ of the universal curve with itself.Comment: 14 pages. AMSLate

The Toda equations and the Gromov-Witten theory of the Riemann sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov-Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found -- the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.Comment: 16 pages, LaTeX2

A Compactification Over M‾g\overline{M}_g Of The Universal Moduli Space of Slope-Semistable Vector Bundles

A projective moduli space of pairs (C,E) where E is a slope- semistable torsion free sheaf of uniform rank on a Deligne- Mumford stable curve C is constructed via G.I.T. There is a natural SL x SL action on the relative Quot scheme over the universal curve of the Hilbert scheme of pluricanonical, genus g curves. The G.I.T. quotient of this product action yields a functorial, compact solution to the moduli problem of pairs (C,E). Basic properties of the moduli space are studied. An alternative approach to the moduli problem of pairs has been suggested by D. Gieseker and I Morrison and completed by L. Caporaso in the rank 1 case. It is shown the contruction given here is isomorphic to Caporaso's compactification in the rank 1 case.Comment: JAMS, to appear, AMSLatex 64 page