Some results on deformations of sections of vector bundles
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- 作者：Abel Castorena ; Gian Pietro Pirola
- 关键词：Continuous rank ; Paracanonical system ; Semi ; regular divisor ; Vanishing theorems
- 刊名：Collectanea Mathematica
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：68
- 期：1
- 页码：9-20
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Applications of Mathematics; Algebra; Geometry; Analysis;
- 出版者：Springer Milan
- ISSN：2038-4815
- 卷排序：68
文摘
Let E be a vector bundle on a smooth complex projective variety X. We study the family of sections \(s_t\in H^0(E\otimes L_t)\) where \(L_t\in Pic^0(X)\) is a family of topologically trivial line bundle and \(L_0={\mathcal {O}}_X,\) that is, we study deformations of \(s=s_0\). By applying the approximation theorem of Artin (Invent Math 5:277–291, 1968) we give a transversality condition that generalizes the semi-regularity of an effective Cartier divisor. Moreover, we obtain another proof of the Severi–Kodaira–Spencer theorem (Bloch In Invent Math 17:51–66, 1972). We apply our results to give a lower bound to the continuous rank of a vector bundle as defined by Miguel Barja (Duke Math J 164(3):541–568, 2015) and a proof of a piece of the generic vanishing theorems (Green and Lazarsfeld, Invent Math 90:389–407, 1987) and (Green and Lazarsfeld, J Am Math Soc 4:87–103, 1991) for the canonical bundle. We extend also to higher dimension a result given in (Mendes-Lopes et al. In Geo Topol 17:1205:1223, 2013) on the base locus of the paracanonical base locus for surfaces.