AbstractsMathematics

Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points

by Lei Song




Institution: University of Illinois – Chicago
Department:
Year: 2014
Keywords: Brill-Noether loci; Semi-regular line bundles; Rational singularities; Hilbert scheme of points on a surface; Universal family; Log canonical threshold
Record ID: 2033503
Full text PDF: http://hdl.handle.net/10027/18980


Abstract

It is well known in algebraic geometry that Hilbert and Picard functors are representable by Hilbert schemes $text{Hilb}(X)$ and Picard schemes $text{Pic}(X)$ respectively. The thesis studies singularities of certain spaces relating to these schemes. It primarily consists of two parts of independent interest. In the first part (Chapter 3), we study the Brill-Noether locus $W^0(X)$ of effective line bundles over a smooth projective variety $X$ of arbitrary dimension; and we show that if a line bundle $L$ is semi-regular, then $W^0(X)$ has rational singularities at $[L]$. Since the semi-regularity holds automatically for all line bundles over a curve, we thereby recover a Kempf's theorem stating that all Brill-Noether loci $W^0_d(C)$ have rational singularities for all smooth projective curve $C$ of genus $g$ and $1le dle g-1$. We also study the local ring $sshf{W^0(X), [L]}$ for such $L$. To show the condition of semi-regularity is not overly strong, we construct a family of examples from ruled surfaces, and make an analysis of one type of components of $W^0_{sr}(X)$. In the second part (Chapter 4), we study the Hilbert scheme of $n$-points on a quasi-projective smooth surface $X$. Specifically, we show that the universal family $Z^n$ over $text{Hilb}^{n}(X)$ has non $mathbb{Q}$-Gorenstein, rational singularities, and its Samuel multiplicity can be described by a quadric in terms of the dimension of socle of zero-dimensional subscheme. In a different but closely related direction, we study the log canonical threshold $c_n$ of the pair $(text{Hilb}^{n}(X), B^n)$, where $X$ is the affine plane and $B^n$ is the exceptional divisor of the Hilbert-Chow morphism, via two approaches. Using the Fulton-MacPherson compactification of configuration spaces and Haiman's work on the $n!$ conjecture, we give a lower bound of $c_n$. On the other hand, by versal deformations of monomial ideals on the plane, we relate $c_n$ to the log canonical threshold of the discriminant of a degree $n$ polynomial in one variable.