Bianca Viray

Hilbert schemes of 8 points

The Hilbert scheme H^d_n of n points in A^d contains an irreducible component R^d_n which generically represents n distinct points in A^d. We show that when n is at most 8, the Hilbert scheme H^d_n is reducible if and only if n = 8 and d >= 4. In the simplest case of reducibility, the component R^4_8 \subset H^4_8 is defined by a single explicit equation which serves as a criterion for deciding whether a given ideal is a limit of distinct points. To understand the components of the Hilbert scheme, we study the closed subschemes of H_n^d which parametrize those ideals which are homogeneous and have a fixed Hilbert function. These subschemes are a special case of multigraded Hilbert schemes, and we describe their components when the colength is at most 8. In particular, we show that the scheme corresponding to the Hilbert function (1,3,2,1) is the minimal reducible example.

An arithmetic intersection formula for denominators of Igusa class polynomials

In this paper we prove an explicit formula for the arithmetic intersection number

On Brauer groups of double covers of ruled surfaces

({\rm CM}(K).{\rm G}_1)_{\ell}

Failure of the Hasse principle for Enriques surfaces

on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus

Failure of the Hasse principle for Châtelet surfaces in characteristic 2

2

Two torsion in the Brauer group of a hyperelliptic curve

curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

({\rm CM}(K).{\rm G}_1)_{\ell}

On Singular Moduli for Arbitrary Discriminants

under strong assumptions on the ramification of the primitive quartic CM field

Evidence for the Dynamical Brauer-Manin Criterion

K

The d-primary Brauer–Manin obstruction for curves

. Yang later proved this conjecture assuming that