Interpolation and Vector Bundles on Curves
Interpolation and Vector Bundles on Curves
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曲线上的插值和向量束
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发表时间：
2014
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通讯作者：
A. Atanasov
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作者：
A. Atanasov
我们定义了曲线上向量束插值的几个概念，并讨论了它们与边坡稳定性的关系。本文的核心内容演示了如何使用退化论证来证明插值。我们使用这些想法来证明度数 $d$ 和亏格 $g$ 的一般连通空间曲线满足 $d geq g+3$ 的插值，除非 $d = 5$ 且 $g = 2$。作为第二个应用，我们证明，当 $d geq 7$、$d geq n+1$ 且 $2d$ 模 $n-1$ 的余数位于 $3$ 和 $n-2$（含）之间时，$mathbb{P}^n$ 中的 $d$ 阶一般椭圆曲线满足弱插值。最后，我们证明插值等价于先验的更严格的强插值概念。如果我们对高维线性空间给出的关联条件感兴趣，这很有用。
We define several notions of interpolation for vector bundles on curves and discuss their relation to slope stability. The heart of the paper demonstrates how to use degeneration arguments to prove interpolation. We use these ideas to show that a general connected space curve of degree $d$ and genus $g$ satisfies interpolation for $d geq g+3$ unless $d = 5$ and $g = 2$. As a second application, we show that a general elliptic curve of degree $d$ in $mathbb{P}^n$ satisfies weak interpolation when $d geq 7$, $d geq n+1$, and the remainder of $2d$ modulo $n-1$ lies between $3$ and $n-2$ inclusive. Finally, we prove that interpolation is equivalent to the---a priori stricter---notion of strong interpolation. This is useful if we are interested in incidence conditions given by higher dimensional linear spaces.