Low dimensional loci and scrolls

Antonio Lanteri, Roberto Munoz, Low dimensional loci and scrolls, Indiana Univ. Math. J. 58 (2009), 2205-2226

Abstract

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Smooth complex polarized varieties $(X, L)$ with a vector subspace
$V \subseteq H^{0} (X, L)$ spanning $L$ are classified under the assumption that the locus $D (X, V)$ of singular elements of $| V |$ has codimension equal to
$dim (X) - i$ , $i = 3, 4, 5$ , the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $dim (X) - 4$ while it is $dim (X) - 3$ if and only if $(X, L)$ is a scroll over a smooth curve. When the codimension is $dim (X) - 5$ and the Picard number is one, only the Plücker embedding of the Grassmannian of lines in $P^{4}$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided.

Antonio Lanteri: Dipartimento di Matematica "F. Enriques"
Università degli Studi di Milano
Via C. Saldini 50
I-20133, Milano, Italy
E-mail address : antonio.lanteri@unimi.it

Roberto Muñoz: Departamento de Matemática Aplicada
Universidad Rey Juan Carlos
Calle Tulipán s/n
28933-Móstoles, Madrid, Spain
E-mail address : roberto.munoz@urjc.es

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