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Determinantal hypersurfaces. (English) Zbl 1076.14534

It is a classical question whether a homogeneous form can be written as a determinant of a matrix of linear forms. The author gives a surprisingly simple general approach to this problem, relating it, via resolutions, to the existence of arithmetically Cohen-Macaulay sheaves on projective hypersurfaces. First, general results about determinantal hypersurfaces are stated, covering also the existence of a special structure (symmetric determinants, Pfaffians). Then, a huge number of applications is given, starting with the case of plane curves, where the determinantal description is related to lines bundles, theta characteristics and rank two bundles. For the respective moduli spaces, several interesting geometric results (like unirationality) are derived. In the surface case, several classical results are recovered and refined; for instance, it is shown that every smooth cubic in \(\mathbb{P}^3\) can be defined by a linear Pfaffian (indeed, such a statement holds for a general surface of degree \(d\) if and only \(d\leq 15\), as shown in the appendix by F.-O. Schreyer by a Macaulay 2 computation). Finally, the case of linear Pfaffian threefolds and fourfolds is treated.

MSC:

14M12 Determinantal varieties
14J70 Hypersurfaces and algebraic geometry

Software:

Macaulay2
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