Alessandro Oneto: The geometry of strength of polynomials
Time: Mon 2019-12-16 15.30
Location: Kräftriket, house 5, room 35
Participating: Alessandro Oneto, Universität Magdeburg
The strength of a homogeneous polynomial f is the smallest number of reducible forms whose sum is equal to f. We call it slice rank if we assume the summands to be only reducible forms that have a linear factor. Normal forms of this type can be very useful to learn geometric properties of the hypersurfaces in projective space: for example, it is immediate to see that if f has slice rank equal to r, then the hypersurface defined by f contains a linear space of codimension r. Similarly, if the hypersurface contains any complete intersection of codimension r, then the strength of f is at most r.
As usual, additive decompositions can be studying by looking at secant varieties of special varieties: in this case, to the varieties of reducible forms. In a recent paper, Catalisano-Geramita-Gimigliano-Harbourne-Migliore-Nagel-Shin, studied the dimensions of these secant varieties by exploiting weak Lefschetz properties of certain artinian algebras and the relation with Fröberg's conjecture. As a consequence of their analysis, they suspect that strength and slice rank coincide for a general form.
The goal of this talk is to explain the relation between these various problems and, after proving some basic properties of these additive decompositions, I will show that the equality between strength and slice rank holds for polynomials in low degree. As a reinterpretation of this result, we deduce bounds on the possible codimensions of complete intersections contained in general hypersurfaces of low degree.
This is based on an ongoing project with Arthur Bik (U. of Bern).