Aaron Bertram

Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the Grothendieck Quot Scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as “Gromov invariants”) on the space of holomorphic maps into Grassmannians. We carry out these computations in the case where the Riemann surface has genus one. Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 E-mail address: bertram@math.utah.edu Department of Mathematics, Brown University, Providence, Rhode Island 02912 E-mail address: daskal@gauss.math.brown.edu Department of Mathematics, University of California, Irvine, California 92717 E-mail address: raw@math.uci.edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

Bridgeland-stable moduli spaces for

It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.It is shown that a Banach space admits an equivalent norm whose modulus of uniform convexity has power-type p if and only if it is Markov p-convex. Counterexamples are constructed to natural questions related to isomorphic uniform convexity of metric spaces, showing in particular that tree metrics fail to have the dichotomy property.

K

Let X be a smooth projective variety over C with the (linearized) action of a complex reductive group G, and let T ⊂ G be a maximal torus. In this setting, there are two geometric invariant theory (GIT) quotients, X//T and X//G, with a rational map Φ : X//T − −> X//G between them. We will further assume that “stable = semistable” in the GIT and that all isotropy of stable points is trivial, so X//T and X//G are smooth projective varieties, and Φ is a G/T fibration.

-trivial surfaces

We apply a conjectured inequality on third chern classes of stable two-term complexes on threefolds to Fujita’s conjecture. More precisely, the inequality is shown to imply a Reider-type theorem in dimension three which in turn implies that KX + 6L is very ample when L is ample, and that 5L is very ample when KX is trivial.

Gromov-witten invariants for abelian and nonabelian quotients

The dth symmetric product of a curve of genus g is a smooth projective variety. This paper is concerned with the little quantum cohomology ring of this variety, that is, the ring having its 3-point Gromov-Witten invariants as structure constants. This is of considerable interest, for example as the base ring of the quantum category in Seiberg-Witten theory. The main results give an explicit, general formula for the quantum product in this ring unless d is in the narrow interval [3/4 g, g-1). Otherwise, they still give a formula modulo third order terms. Explicit generators and relations are also given unless d is in [4/5 g - 3/5, g-1). The virtual class on the space of stable maps plays a significant role. But the central ideas ultimately come from Brill-Noether theory: specifically a formula of Harris-Tu for the Chern numbers of determinantal varieties. The case d = g-1 is especially interesting: it resembles that of a Calabi-Yau 3-fold, and the Aspinwall-Morrison formula enters the calculations. A detailed analogy with Giventals work is also explained.

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

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.