808 research outputs found

### 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

### 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

### 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

### 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

### Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable
maps to projective space is considered. Generators and Picard numbers are
computed. A recursive algorithm computing all top intersection products of
Q-Divisors is established. As a corollary, an algorithm computing all
characteristic numbers of rational curves in P^r is proven (including simple
tangency conditions). Computations of these characteristic numbers are carried
out in many examples. The degree of the 1-cuspidal rational locus in the linear
system of degree d plane curves is explicitly evaluated.Comment: AMSLaTex 31 page

### A Compactification Over $\overline{M}_g$ 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

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