GIT and singularities on algebraic surfaces

Abstract

One of main problems in algebraic geometry is to construct moduli spaces that parameterize certain algebraic objects. And Geometric Invariant Theory (GIT) is one of the oldest approaches to moduli problems. For moduli spaces to have nice properties, one usually has to discard certain `unstable' objects. The machinery of GIT gives powerful general methods in turning the collection of stable objects into a projective moduli space. However, the definition of stability it uses can be hard to check, as it involves looking at all degenerations of a given object. Thus, because of its generality, it leaves task of identifying which objects are stable precisely.

For the moduli space of curves one can include any irreducible curve which has at worst ordinary double point singularities, but it remains unclear what happens in higher dimensions. In this thesis, we study the GIT stability and singularities on certain algebraic surfaces.

Firstly, we consider hypersurface sections of quadric threefolds. Let $S$ be a complete intersection of a smooth quadric threefold and a hypersurface of degree $d$ in $\mathbb P^4$. We analyze GIT stability of $S$ with respect to the natural $G=SO(5, \mathbb C)$-action, and understand the type of singularities when $S$ is not stable or unstable by using GIT stability analysis. In particular, we prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.

Secondly, we consider nets of quadrics in $\mathbb{P}^5$ and associated discriminants. Let $S$ be a complete intersection surface defined by a net $\Lambda$ of quadrics in $\mathbb P^5$. We analyze GIT stability of nets of quadrics in $\mathbb P^5$ up to projective equivalence, and discuss some connections between a net of quadrics and the associated discriminant sextic curve. In particular, we prove that if $S$ is normal and the discriminant $\Delta(S)$ of $S$ is stable then $\Lambda$ is stable. And we prove that if $S$ has the reduced discriminant and $\Delta(S)$ is stable then $\Lambda$ is stable. Moreover, we prove that if $S$ has simple singularities then $\Delta(S)$ has simple singularities.