Algebraic Geometry Seminar
Yank\i\ Lekili
UIC
A modular compactification of $\mathcal{M}_{1,n}$ from $A_{\infty}$-structures
Abstract: Motivated by homological mirror symmetry, for each n, we define a certain
finite dimensional graded associative algebra $E_n$ and study the moduli
space of minimal $A_\infty$ structures on $E_n$. Surprisingly, we can identify
this moduli space with a modular compactification (due to Smyth) of the
moduli of curves of genus 1 with $n$ marked points. The corresponding moduli
stack, denoted by $\mathcal{M}_{1,n}(n-1)$, is a projective irreducible DM-stack. Our
realization of this space gives a description of these moduli spaces and
the universal curves over them by explicit equations. This enables us to
discover various geometric properties of $\mathcal{M}_{1,n}(n-1)$. For example, we prove
that they are normal and Gorenstein, show that their Picard groups have no
torsion and that they have rational singularities if and only if $n \leq 11$.
The case of $n=1$ and the algebra $E_1$ was studied earlier by the speaker and
Perutz, the current report is for $n>1$ on a joint work with A. Polishchuk.
Wednesday September 10, 2014 at 4:00 PM in SEO 427