Iterated sparse discriminants and singular intersections of hypersurfaces

E. KUNZ, WITH THE ASSISTANCE OF AND CONTRIBUTIONS BY DAVID A. COX AND ALICIA DICKENSTEIN

Seminario; Seminario de Geometría Algebraica; 2019

It it wellknown that two generic quadric surfaces intersect in a nonsingular quarticspace curve, but when the intersection is not transverse this intersectioncurve may degenerate to a finite number of different possible types of singularcurves.  In the nice paper by Farouki et al. (1989), the authors formulatea way of computing the condition for a degenerate intersection in this case,which refines in the real case and with an algorithmic point of view aclassical treatise by  Bromwich (1906). Independently, Schläfli (1953)studied the degenerate intersection of two hypersurfaces described bymultilinear equations.In joint work with S. di Rocco and R. Morrison, we present a general framework of iterated sparse discriminants to characterizethe singular intersection of hypersurfaces with a given monomial support A, whichgeneralizes both previous situations. We  study the connection of iterateddiscriminants with the notion of mixed discriminant and the singularities ofthe sparse discriminant associated to A.