Abstract
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.
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Acknowledgments
The authors give thanks to Rita Pardini and Miguel Angel Barja for fruitful discussions and suggestions. Special thanks are due to the referee for valuable comments which helped us to improve the manuscript. The first named author give thanks to the Department of Mathematics of the University of Pavia for their warm hospitality and the use of their resources during a sabbatic year.
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The first named author was partially supported by sabbatical fellowships from CONACyT-México (research grant 133228); DGAPA-UNAM (México) and was partially supported as Visiting Professor by INdAM (GNSAGA).
The second named author is partially supported by INdAM (GNSAGA); PRIN 2012 “Moduli, strutture geometriche e loro applicazioni” and FAR 2014 (PV) “Varietà algebriche, calcolo algebrico, grafi orientati e topologici”.
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Castorena, A., Pirola, G.P. Some results on deformations of sections of vector bundles. Collect. Math. 68, 9–20 (2017). https://doi.org/10.1007/s13348-016-0169-z
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DOI: https://doi.org/10.1007/s13348-016-0169-z