Residual intersections and Schubert varieties
Sara Angela Filippini
15-06-2023 - 14:15
Largo San Leonardo Murialdo, 1
The notion of residual intersections was introduced by Artin and Nagata. Roughly speaking, given an algebraic variety $X$ and a closed subscheme $Y$ in $X$, which is contained in another closed subscheme $Z$, then a closed subscheme $W$ such that $W cup Y = Z$ is a residual intersection of $Y$ in $Z$.
This idea can be formalized as follows: Let $I$ be an ideal in a local Cohen-Macaulay ring $R$, and $A = (a_1, ldots, a_s) subsetneq I$. Then $J = A:I$ is called an s-residual intersection of $I$ if $ht(J) geq s geq ht(I)$. Residual intersections provide a generalization of linkage. Indeed, if $J = A:I$ and $I = A:J$ for $A$ a regular sequence, $I$ and $J$ are said to be linked.
I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard deteminantal ideals and Pfaffian ideals respectively arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with J. Torres and J. Weyman.
org: Turchet Amos