Patch ideals and Peterson varieties

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.