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Brendan Hassett

Publications and source records attributed to Brendan Hassett.

62 records · Page 4Linked to original sources
arXiv2003

Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on $¶^1$, which is isomorphic to a certain moduli space of stable maps to $¶^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of $\SL_2$ in the space of binary forms of degree $n\ge 3$.

math.AG↗
arXiv2002

Moduli spaces of weighted pointed stable curves

A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sum of the corresponding weights is no greater than one. We construct moduli spaces for these objects using methods of the log minimal model program, and describe the induced birational morphisms between moduli spaces as the weights are varied. In the genus zero case, we explain the connection to Geometric Invariant Theory quotients of points in the projective line, and to compactifications of moduli spaces studied by Kapranov, Keel, and Losev-Manin.

math.AG↗
arXiv1999

Abelian fibrations and rational points on symmetric products

Given a variety over a number field, are its rational points potentially dense, i.e., does there exist a finite extension over which rational points are Zariski dense? We study the question of potential density for symmetric products of surfaces. Contrary to the situation for curves, rational points are not necessarily potentially dense on a sufficiently high symmetric product. Our main result is that rational points are potentially dense for the Nth symmetric product of a K3 surface, where N is explicitly determined by the geometry of the surface. The basic construction is that for some N, the Nth symmetric power of a K3 surface is birational to an abelian fibration over P^N. It is an interesting geometric problem to find the smallest N with this property.

math.AG↗
arXiv1999

Geometry of equivariant compactifications of $G^n_a$

Equivariant compactifications of reductive groups can be described by combinatorial data. On the other hand, equivariant compactifications of the additive group G^n_a are more complicated in at least two respects. First, they often admit moduli. Second, even simple varieties (like projective spaces) admit many different structures as equivariant compactifications of the additive group. We give a dictionary relating Artinian local rings, certain systems of partial differential equations, and equivariant compactifications of G^n_a. As an application, we classify the structures on varieties like projective spaces, ruled surfaces, and certain threefolds. We consider these results as a first step in a systematic study of G^n_a-equivariant birational geometry.

math.AG↗
arXiv1995

Correlation for Surfaces of General Type

The main geometric result of this paper is that given any family of surfaces of general type f:X-->B, for sufficiently large n the fiber product X^n_B dominates a variety of general type. This result is especially interesting when it is combined with Lang's Conjecture. This states that for a variety V of general type over a number field K, the K rational points V(K) are not Zariski dense in V. Assuming Lang's Conjecture, we prove the existence of a uniform bound on the degree of the Zariski closure of the K-rational points of a surface of general type.

alg-geom↗