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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
##### Keywords:
resultants; discriminant; Chow form; Tate resolution; Ulrich bundles
##### Software:
CubicSurfaces; DetRepOfCubicDiscriminant.m2; GitHub; Macaulay2
Full Text:
##### References:
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