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A Degree Formula for Secant Varieties of Curves

Published online by Cambridge University Press:  21 August 2013

Rüdiger Achilles ,
Mirella Manaresi  and
Peter Schenzel
Rüdiger Achilles
Affiliation:
Dipartimento de Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, ([email protected]; [email protected])
Mirella Manaresi
Affiliation:
Dipartimento de Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy, ([email protected]; [email protected])
Peter Schenzel
Affiliation:
Institut für Informatik, Martin-Luther-Universität, Von-Seckendorff-Platz 1, 06120 Halle (Saale), Germany, ([email protected])
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Abstract

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Using the Stückrad–Vogel self-intersection cycle of an irreducible and reduced curve in pro-jective space, we obtain a formula that relates the degree of the secant variety, the degree and the genus of the curve and the self-intersection numbers, the multiplicities and the number of branches of the curve at its singular points. From this formula we deduce an expression for the difference between the genera of the curve. This result shows that the self-intersection multiplicity of a curve in projective N-space at a singular point is a natural generalization of the intersection multiplicity of a plane curve with its generic polar curve. In this approach, the degree of the secant variety (up to a factor 2), the self-intersection numbers and the multiplicities of the singular points are leading coefficients of a bivariate Hilbert polynomial, which can be computed by computer algebra systems.

Keywords

curves in projective spaceself-intersection multiplicitiesarithmetic genusgeometric genussecant varietiesPrimary 14C1714E22Secondary 14H2014N15
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 57 , Issue 2 , June 2014 , pp. 305 - 322
Copyright
Copyright © Edinburgh Mathematical Society 2014 

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