论文标题
线束的同时在发病率对应关系上
Cohomology of line bundles on the incidence correspondence
论文作者
论文摘要
对于尺寸n的有限维矢量空间V，我们考虑p（v）x p（v*）中的发病率对应关系（或部分标志品种）x，参数对，由一个点和包含它的超平面组成。我们完全表征了特征性p> 0中x上线束的共同体学组的消失和非变化行为。如果n = 3，则x是V的完整标志品种，并且表征包含在70年代的格里菲斯（Griffith）论文中。在特征0中，Borel-Weil-Bott Theorem描述了所有V的共同体学组。我们的策略是在计算（曲折）在投射空间上分裂的权力的（曲折的曲线）的计算协同学来重述问题，然后我们使用Frobenius引起的自然截断进行了研究，并仔细估计了Castelnuovo-Mumumford的规律性。当n = 3时，我们从Linyuan Liu的最新作品中恢复了字符的递归描述，而对于一般n，我们为限制的线束集合的共同体提供了字符公式。我们的结果表明，被截断的Schur功能是共同体特征的自然构件。
For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X in P(V) x P(V*), parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel-Weil-Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo-Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.
