The variety of circular complexes and \(F\)-splitting. (English) Zbl 0955.14036
Let \(V_0,V_1\) be two finite-dimensional vector spaces over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(L=\operatorname{Hom} (V_0,V_1) \times\operatorname{Hom} (V_1,V_0)\) and let \(H=\text{GL}(V_0) \times\text{GL}(V_1)\). Then \(H\) acts on \(L\) by \((g_1,g_2) \cdot(f_1,f_2) =(g_2f_1g_1^{-1}, g_1f_2g_2^{-1})\).
An element \(f=(f_1,f_2)\in L\) is called a circular complex if \(f_1\circ f_2= f_2 \circ f_1=0\).
The main theorem of the present paper asserts that the \(H\)-orbit closure \(\overline O_f\) of a circular complex \(f\) is normal, Cohen-Macaulay with rational singularities. It may be recalled that Strickland proved that each component of the full variety of the circular complexes in \(L\) is normal and Cohen-Macaulay.
An element \(f=(f_1,f_2)\in L\) is called a circular complex if \(f_1\circ f_2= f_2 \circ f_1=0\).
The main theorem of the present paper asserts that the \(H\)-orbit closure \(\overline O_f\) of a circular complex \(f\) is normal, Cohen-Macaulay with rational singularities. It may be recalled that Strickland proved that each component of the full variety of the circular complexes in \(L\) is normal and Cohen-Macaulay.
Reviewer: Shrawan Kumar (Chapel Hill)
MSC:
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14G15 | Finite ground fields in algebraic geometry |
