Iterated discriminants and singular space curves

A DICKENSTEIN

Barcelona

Otro; ARCADES Doctoral School II and ESR Days in Barcelona; 2018

Universidad de Barcelona

In general, twoquadric surfaces intersect in a nonsingular quartic space curve, but underspecial circumstances this intersection curve may degenerate to a finite numberof different possible types of singular curves. These degenerate space curvesare important since they occur frequently in practice and, unlike the genericcase, they admit rationalparameterizations. In the nice paper [3], the authors formulate the conditionfor a degenerate intersection, which refines the study of the real case andwith an algorithmic point of view the classical treatise [1]. Independently,the condition for a degenerate intersection of two surfaces of tensor type (ormore generally, of two hypersurfaces described by multilinear equations) isstudied in [4].Folllowing jointwork with S. di Rocco and R. Morrison in [2], I will present a generalframework of iterated discriminants to characterize the singular intersectionof hypersurfaces with a given monomial support, which generalizes both previoussituations. I will explain the notion of mixed discriminant and the relationwith these iterated discriminants.[1] T. J. I?A.Bromwich: Quadratic forms and their classification by means of invariant-factors.Cambridge Univ. Press, Cambridge, Jbuch 37, 1906.[2] A. Dickenstein,S. di Rocco, R. Morrison: Iterated multivariate discriminants and mixeddiscriminants, Manuscript, 2018.[3] R.T. Farouki,C.A. Neff, M.A. O?Connor: Automatic parsing of degenerate quadricsurfaceintersections, ACM Transactions on Graphics 8 (3) (1989) 174-203.[4] L. Schläfli:Gesammelte mathematische Abhandlungen. Band II, Verlag Birkhäuser, Basel, 1953.