Moduli Space of A-Infinity Structures and Nonreduced Curves of Genus 0
Datum
2022-10-04
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University of Oregon
Zusammenfassung
In this thesis, we study $A_\infty$-structures arising from derived categories of certain algebraic curves. More precisely, we consider pairs $(\mathcal{O}_C,\mathcal{O}_D)$, where $C$ is an irreducible projective curve over a field $k$ with $H^0(C,\mathcal{O}_C)=k$ and $H^1(C,\mathcal{O}_C)=0$, and $D\sub C$ is a Cartier divisor of degree $2$, supported at one point. They satisfy certain categorical properties encoded in the notion of an $R$-pair (of genus $0$), $(E,F)$, which we will define. In particular, $E$ is exceptional and $F$ is $R$-spherical which is a version of the notion of a $1$-spherical object defined in the work of Seidel and Thomas. The main result of this thesis is to prove the equivalence between the moduli of the $R$-pairs and that of certain filtered algebras which permit a simpler description, i.e. given by the quotient stack of a closed subscheme of $\mathbb{A}^3$ for some action of $\mathbb{G}_a$.