1999/01/20 by Luca Chiantini, L. Chiantini, A. F. Lopez +6
Computer Science · Mathematics · #14C99 #14J70 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation #math.AG #msc:14C99 #msc:14J70
paper · pdf · doi:10.48550/arxiv.math/9901083
We made some improvements in the introduction and definitions. In an effort to clarify the arguments we separated the 1-filling case from the r-filling case and we gave a more detailed proof of the key lemma. The article will appear in the Math. Proc. Cambridge Philos. Soc
openalex publication_date 1999/01/20 · arxiv created 2001/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let W be a projective variety of dimension n+1, L a free line bundle on W, X in H0(Ld) a hypersurface of degree d which is generic among those given by sums of monomials from L, and let f : Y → X be a generically finite map from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X in H0(Ld), f deforms in a family such that the corresponding deformations of Yr dominate Wr. Under these hypotheses we give a lower bound for the dimension of a certain linear system on the Cartesian product Yr having certain vanishing order on a diagonal locus as well as on a double point locus. This yields as one application a lower bound on the dimension of the linear system |KY - (d - n + m)f^*L - f^*KW| which generalizes results of Ein and Xu (and in weaker form, Voisin). As another perhaps more surprising application, we conclude a lower bound on the number of quadrics containing certain projective images of Y.