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Determinantal representations of the cubic discriminant. (English) Zbl 1451.14167
Summary: We compute and study two determinantal representations of the discriminant of a cubic quaternary form. The first representation is the Chow form of the 2-uple embedding of \(\mathbb{P}^3\) and is computed as the Pfaffian of the Chow form of a rank 2 Ulrich bundle on this Veronese variety. We then consider the determinantal representation described by [E. J. Nanson, Proc. R. Soc. Edinburgh 22, 353–358 (1899; JFM 30.0161.09)]. We investigate the geometric nature of cubic surfaces whose discriminant matrices satisfy certain rank conditions. As a special case of interest, we use certain minors of this matrix to suggest equations vanishing on the locus of \(k\)-nodal cubic surfaces.
MSC:
14Q10 Computational aspects of algebraic surfaces
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14M12 Determinantal varieties
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