Ranks of cubic surfaces

Anna Seigal (Oxford University)

Monday 27th January 16:00-17:00 Maths 311B

Abstract

The symmetric rank of a polynomial is its shortest expression as a sum of powers of linear forms. The rank is the non-symmetric analogue that arises from viewing the polynomial as a symmetric tensor. In this talk, we study the non-symmetric rank and the symmetric rank of cubic surfaces, and show that the two notions coincide over the complex numbers. The results extend to order three tensors of all sizes, implying the equality of rank and symmetric rank when the symmetric rank is at most seven. We then explore the connection between the rank of a cubic surface and the singularities of its vanishing locus. Cubic surfaces with isolated singularities are known to fall into 22 singularity types. We compute the rank of a cubic surface of each singularity type. This enables us to find the possible singular loci of a cubic surface of fixed rank. Finally, we study connections to the Hessian discriminant. We show that a singular cubic surface lies on the Hessian discriminant unless all singularities are ordinary double points, and we show that the Hessian discriminant is the closure of the rank six cubic surfaces. This is based on joint work with Eunice Sukarto.

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