Department of

October 2016 November 2016 December 2016 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 1 2 3 4 5 1 2 3 2 3 4 5 6 7 8 6 7 8 9 10 11 12 4 5 6 7 8 9 10 9 10 11 12 13 14 15 13 14 15 16 17 18 19 11 12 13 14 15 16 17 16 17 18 19 20 21 22 20 21 22 23 24 25 26 18 19 20 21 22 23 24 23 24 25 26 27 28 29 27 28 29 30 25 26 27 28 29 30 31 30 31

Friday, November 4, 2016

**Abstract:** Given a projective variety X with an action of a complex reductive group G, the quotient space $X/ G$ may not exist in the category of algebraic varieties. In order to fix this problem, Geometric Invariant Theory gives a construction of a $G$-invariant open subset $X^{ss}$ of X for which the algebraic quotient exists. The Kirwan-Ness (KN) stratification refines $X$ and its unique open stratum coincides with the set $X^{ss}$. When $X$ is a linear representation $G$, the semistable locus $X^{ss}\subset X$ and a KN stratification of $X \backslash X^{ss}$ are associated to a choice of a homomorphism $G\rightarrow \mathbb{G}_m$ . That is, we can write $X=X^{ss}\sqcup \bigsqcup_{\alpha\in \text{KN}}S_\alpha,$ where $S_{\alpha}$ are locally closed smooth pieces and KN indexes the 1-parameter subgroups that determine the stratification. In this situation, Nevins and McGerty give an algorithm to find the KN 1-parameter subgroups. I will ilustrate the algorithm with some examples and at the end I will focus on the case when $X$ is a representation of the cyclic quiver.