Determinantal representations of the cubic discriminant. (English) Zbl 1451.14167

Bunnett, Dominic; Keneshlou, Hanieh

Matematiche 75, No. 2, 489-505 (2020).

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

Citations:

JFM 30.0161.09

Software:

Macaulay2; CubicSurfaces; DetRepOfCubicDiscriminant.m2; GitHub

References:

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