Local properties of secant varieties in symmetric products. II. (English) Zbl 0701.14046
Let C be a smooth curve, \(L\in Pic(C),V\subseteq H^ 0(C,L)\) defining a morphism into \({\mathbb{P}}^ n\); there is a bundle \(E_ L\) on the d-th symmetric product \(C^{(d)}\) and a map \(\sigma:\;V\otimes {\mathcal O}_{C^{(d)}}\to E_ L\) whose degeneracy locus gives the secant subschemes \(V^ n_ d\) of \(C^{(d)}\). In part I of this paper by H. Huibregtse and the author (cf. the preceding review) it is given a local matrix description of \(\sigma\). Here the results and methods of part I are applied to compute several examples: local tangent space dimensions of \(V^ 1_ d\), tangent cones of \(V^ 1_ n\), stationary bisecants for space curves. In a few cases the painstaking explicit calculations are omitted.
Reviewer: E.Ballico
MSC:
14M15 | Grassmannians, Schubert varieties, flag manifolds |
14H45 | Special algebraic curves and curves of low genus |
14B12 | Local deformation theory, Artin approximation, etc. |
14N05 | Projective techniques in algebraic geometry |