曲线的半稳定性和Hilbert-Kunz重数
Semistability and Hilbert-Kunz multiplicities for curves
摘要：
研究了正特征非奇异曲线的Hilbert-Kunz重数。我们分析了曲线及其丰富线丛的核层的Frobenius半稳定性与HK重数之间的关系。这将导致一个下界，实现当且仅当核心层是Frobenius半稳定的，否则公式的HK多重性的参数测量失败的Frobenius半稳定。作为副产品，给出曲线上的向量丛的显式示例，其$n$次迭代Frobenius拉回不是半稳定的，而其$(n-1)$次这样的拉回是半稳定的，其中$n>0$是任意的。
英文摘要：
We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its $(n-1)$-th such pullback is semistable, where $n>0$ is arbitrary.
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