Invariants of the singularities of secant varieties of curves

Daniel Brogan
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Abstract

Consider a smooth projective curve and a given embedding into projective space via a sufficiently positive line bundle. We can form the secant variety of $k$-planes through the curve. These are singular varieties, with each secant variety being singular along the last. We study invariants of the singularities for these varieties. In the case of an arbitrary curve, we compute the intersection cohomology in terms of the cohomology of the curve. We then turn our attention to rational normal curves. In this setting, we prove that all of the secant varieties are rational homology manifolds, meaning their singular cohomology satisfies Poincar\'e duality. We then compute the nearby and vanishing cycles for the largest nontrivial secant variety, which is a projective hypersurface.
曲线正割品种奇点的不变式
考虑一条光滑的投影曲线和通过足够正的线束嵌入投影空间的给定嵌入。我们可以形成穿过曲线的 $k$-planes 的 secant variety。这些平面都是奇异平面,每个奇异平面沿着最后一个平面都是奇异的。我们将研究这些奇点的不变式。在任意曲线的情况下,我们用曲线的同调来计算交点同调。然后,我们将注意力转向有理正态曲线。在这种情况下,我们证明所有的正割曲线都是有理同调流形,这意味着它们的奇异同调满足 Poincar\'e 对偶性。然后,我们计算了最大的非难secant varieties的邻近循环和消失循环,它是一个投影超曲面。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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