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Quartic 3-fold: Pfaffians, instantons and half-canonical curves

A. Iliev·D. Markushevich
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Abstract

A generic quartic 3-fold X admits a 7-dimensional family of representations as the Pfaffian of an 8 by 8 skew-symmetric matrix of linear forms. This provides a 7-dimensional moduli space M of rank 2 vector bundles on X. A precise geometric description of a 14-dimensional family of half-canonical curves C of genus 15 in X such that the above vector bundles are obtained by Serre's construction from C is given. It is proved that the Abel-Jacobi map of this family factors through M, and the resulting map from M to the intermediate Jacobian is quasi-finite. In particular, every component of M has non-negative Kodaira dimension. Some other constructions of rank 2 vector bundles with small Chern classes are discussed; it is proved that the smallest possible charge of an instanton on X is 4.

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A. Iliev, D. Markushevich. 1999-11-20. Quartic 3-fold: Pfaffians, instantons and half-canonical curves. https://arxiv.org/abs/math/9910133

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