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Patch ideals and Peterson varieties. (English) Zbl 1267.14066

The authors describe an instrument, the patch ideals, to study varieties \(X\) in the complete flag variety over \(\mathbb{C}\): the patch ideals encode neighbourhoods of \(X\).
They study mainly the Peterson varieties \(\mathrm{Pet}_n\), namely the variety of flags \(0\subset F_1\subset \dots \subset F_n\) such that \(N\cdot F_i\subset F_{i+1}\), where \(N\) is a arbitrarily fixed regular nilpotent matrix. They give coordinates and equations that scheme-theoretically cut out a natural open neighbourhood (the patch) of any point of the \(\mathrm{Pet}_n\). The ideal generated by these equations (the patch ideal) is radical; moreover it is complete intersection and hence Cohen-Macaulay. Thus, the zero scheme is reduced if it is generically reduced. To prove the latter, they establish the existence of a smooth point. Consequently, \(\mathrm{Pet}_n\) is a local complete intersection, and therefore Cohen-Macaulay and Gorenstein.
Consequently, they combinatorially describe the singular locus of the \(\mathrm{Pet}_n\) (using a decomposition of \(\mathrm{Pet}_n\) into affine cells), give an explicit equivariant \(K\)-theory localization formula and extend some results of Kostant and of Peterson to intersections of Peterson varieties with Schubert varieties.
Finally, they use patch ideals to briefly analyze other examples of torus invariant subvarieties of the complete flag variety, including Richardson varieties and Springer fibers.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14M10 Complete intersections
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