Patch ideals and Peterson varieties

TL;DR

This paper studies the algebraic structure of Peterson varieties using patch ideals, revealing their complete intersection property, describing their singular loci, and extending known results to intersections with Schubert varieties.

Contribution

It determines generators for patch ideals of Peterson varieties, proves they are complete intersections, and extends results to related subvarieties, providing new algebraic and geometric insights.

Findings

Patch ideals of Peterson varieties are complete intersections.

Singular loci of Peterson varieties are explicitly described.

Extended results to intersections with Schubert varieties.

Abstract

Patch ideals encode neighbourhoods of a variety in GL_n/B. For Peterson varieties we determine generators for these ideals and show they are complete intersections, and thus Cohen-Macaulay and Gorenstein. Consequently, we combinatorially describe the singular locus of the Peterson variety; give an explicit equivariant K-theory localization formula; and extend some results of [B. Kostant '96] and of D. Peterson to intersections of Peterson varieties with Schubert varieties. We conjecture that the projectivized tangent cones are Cohen-Macaulay, and that their h-polynomials are nonnegative and upper-semicontinuous. Similarly, we use patch ideals to briefly analyze other examples of torus invariant subvarieties of GL_n/B, including Richardson varieties and Springer fibers.