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Order 1 Congruences of Lines with smooth Fundamental Scheme
Peskine, Christian
2015
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e-ISSN
2464-8728
Abstract
In this note we present a notion of fundamental scheme for
Cohen-Macaulay, order I, irreducible congruences of lines. We show
that such a congruence is formed by the k-secant lines to its fundamental
scheme for a number k that we call the secant, index of the congruence.
if the fundamental scheme X is a smooth connected variety in FN, then
k = (N — l)/(c — 1) (where c is the codimension of X) and X comes
equipped with a special tangency divisor cut out by a virtual hypersurface of degree k — 2 (to be precise, linearly equivalent to a section by
an hypersurface of degree (k — 2) without being cut by one). This is
explained in the main theorem of this paper. This theorem is followed
by a complete classification of known locally Cohen-Macaulay order 1
congruences of lines with smooth fundamental scheme. To conclude we
remark that according to Zak’s classification of Severi Varieties and
Hartshome conjecture for low codimension varieties, this classification
is complete.
Series
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
47 (2015)
Subjects
Publisher
EUT Edizioni Università di Trieste
Languages
en
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