The blowing up Horace method: Application to interpolation in degree four
Here it is proved the following theorem which completes the case of polynomials of degree \(d \geq 5\) proved by \textit{J. Alexander} [Compos. Math. 68, No. 3, 305-354 (1988; Zbl 0675.14025)]. Theorem A: In the vector space of homogeneous polynomials of degree \(\leq 4\) in \(n\) variables over an infinite field the subspace formed by the polynomials vanishing together with their first derivatives at \(r\) sufficiently general points \(P_ 1,\ldots,P_ r\) has codimension \(r(n+1)\) (i.e. the expected one). -- The main point of the paper is the proof which shows how strong for these problems the many versions of Horace method are in the hands of the authors. Here the authors introduce and use a variant in which the inductive step is carried over a blowing up of the original variety \(X\) \((\mathbb{P}^ n\) for theorem A) on which one wants to prove a theorem on the cohomology groups of the general element of a family of subschemes of \(X\).
scientific article; zbMATH DE number 39402