Seminar
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Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Finite F-representation type for homogeneous coordinate rings
Given a variety X over a field of characteristic p and an ample line bundle L on X, one natural question is whether the homogeneous coordinate ring R(X,L) has finite F-representation type (FFRT). While the case of curves is well understood, there are few known examples of varieties with or without this property in higher dimensions. We prove that a large class of homogeneous coordinate rings (essentially, those of Calabi–Yau or general-type varieties) will fail to have finite F-representation type, via an analysis of their rings of differential operators. This illustrates a connection between the commutative-algebraic property of FFRT, and the algebro-geometric properties of positivity/negativity of tangent sheaves of varieties. This also provides instructive examples of the structure of the ring of differential operators for non-F-pure rings, which to this point have largely been unexplored. We will also discuss the case of Fano varieties: Recent work has provided non-toric smooth Fano varieties that do have FFRT (Grassmannians Gr(2,n) and the quintic del Pezzo surface). However, it seems unlikely that this will be true for all Fano varieties; we will present conjectural evidence that “in general” smooth Fano varieties will often fail to have FFRT.
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