Elementary Transformations of Pfaffian Representations of Plane Curves

Details Elementary Transformations of Pfaffian Representations of Plane Curves Elementary Transformations of Pfaffian Representations of Plane Curves

2009-08-25

arxiv.org/abs/0908.3554v1

Mathematics

Algebraic Geometry

24 pages, 1 figure

Scientific paper

Let $C$ be a smooth curve in $\PP^2$ given by an equation F=0 of degree $d$. In this paper we consider elementary transformations of linear pfaffian representations of $C$. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on $C$ with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of $C$ can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.

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