Gordan ideals in the theory of binary forms
The author considers subvarieties \(X_{p,q}\) associated with a two dimensional complex vector space \(V\) and binary forms of degree \(g\) having a root of multiplicity \(\geq p\). He started the study of these objects in J. Algebra, 122, No. 1, 244-249 (1989; Zbl 0689.14001). In this paper he finds an explicit deformation of the rings of covariants of subvarieties \(X_{p,g}\) to a ring generated by monomials. There is deduced an explicit estimate for the degrees of generating covariants. It is proved that if \(2k - 2 < g/2\) then the defining ideal \(J_{g - k + 1}\) of \(X_{g - k + 1,g}\) is generated by quadrics. This allows a geometric interpretation of Gordan's algorithm for computing covariants of binary forms [cf. \textit{J. H. Grace} and \textit{A. Young}, ``The algebra of invariants'' (1903)]. Finally there are obtained applications of multiplicity ideals to find the invariants of binary forms of low degrees. The paper contains many suggestive examples.
deformation of the rings of covariants of subvarieties
scientific article; zbMATH DE number 554248