An infinitesimal study of secant space divisors. Appendix by Trygve Johnsen
The author studies \(e\)-secants to a smooth irreducible complete curve \(C\) defined over an algebraically closed field, considered as points in the \(e\)'th symmetric product \(C^{(e)}\) of \(C\). Such a secant is called an \((e-f-1)\)--space divisor for a linear system \(g_d^n\) if \(n\geq e-f\) and \(\text{dim}[g_d^n(D)] =n-e-f\). When \(g_d^n\) defines an embedding into a projective space \(\mathbb{P}^n\) we have that, when \(D\) is an \((e-f-1)\)--space divisor for \(g_d^n\), then the points of \(D\) span a subspace \(\langle D\rangle\) of \(\mathbb{P}^n\) of dimension \((e-f-1)\), and the intersection \(\langle D\rangle \cap C\) in \(\mathbb{P}^n\) will have length at least \(D\). In the article under review very precise information is given about the tangent space of the subscheme \(V_{e}^{e-f}\) of \(C^{(e)}\) consisting of \(e\)-secant \((e-f-1)\)--space divisors. The dimension is determined, thus answering a question by the author. Several of the results are similar to results of A. Laudal, in a slightly different setting. In an appendix T. Johnson gives a natural description for some of the results in terms of tangent spaces of natural subschemes of grassmannians.
scientific article; zbMATH DE number 952352
scientific article