论文标题
向量束的矢量束的一般张量产物的共同体学
The cohomology of general tensor products of vector bundles on the projective plane
论文作者
论文摘要
计算两个矢量束的张量产物的共同体学是对其模量空间的研究以及在表示理论，组合和物理学的应用中。这些计算在Brill-Noether Loci，Birational Gealtry几何和$ S $ duality的构建中起着基本作用。利用$ \ mathbb {p}^2 $上的Moduli带束带的模型空间的最小模型程序的最新进展，我们计算了$ \ mathbb {p}^2 $的一般半固定捆的张量产品的共同体。这解决了多项式插值问题的自然更高等级的概括。更确切地说，让$ v $和$ w $是$ \ mathbb {p}^2 $上的两个稳定捆绑包的Chern字符，并假设$ W $取决于$ V $。令$ v \ in m（v）$和m（w）$中的$ w \是两个通用稳定捆绑包。我们完全计算了$ v \ otimes w $的共同体。特别是，我们表明，如果$ w $很出色，则$ v \ otimes w $最多具有一个由坡度和欧拉（Euler）特征确定的非零共同体学组，并将Drézet，Göttsche和Hirschowitz的基础结果推广。我们表征了$ m（v）$上有效的Brill-Noether除数的不变性。我们还表征了$ v \ otimes w $全球生成的何时。鉴于模量空间的生育几何形状，我们的计算是规范的，这表明在其他表面上解决类似问题的路线图。
Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the construction of Brill-Noether loci, birational geometry and $S$-duality. Using recent advances in the Minimal Model Program for moduli spaces of sheaves on $\mathbb{P}^2$, we compute the cohomology of the tensor product of general semistable bundles on $\mathbb{P}^2$. This solves a natural higher rank generalization of the polynomial interpolation problem. More precisely, let $v$ and $w$ be two Chern characters of stable bundles on $\mathbb{P}^2$ and assume that $w$ is sufficiently divisible depending on $v$. Let $V \in M(v)$ and $W \in M(w)$ be two general stable bundles. We fully compute the cohomology of $V \otimes W$. In particular, we show that if $W$ is exceptional, then $V \otimes W$ has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We characterize the invariants of effective Brill-Noether divisors on $M(v)$. We also characterize when $V\otimes W$ is globally generated. Our computation is canonical given the birational geometry of the moduli space, suggesting a roadmap for tackling analogous problems on other surfaces.
