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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 18, Issue 4

Issues

Rational curves on Del Pezzo manifolds

Adrian Zahariuc
Published Online: 2018-07-20 | DOI: https://doi.org/10.1515/advgeom-2018-0010

Abstract

We exploit an elementary specialization technique to study rational curves on Fano varieties of index one less than their dimension, known as del Pezzo manifolds. First, we study the splitting type of the normal bundles of the rational curves. Second, we prove a simple formula relating the number of rational curves passing through a suitable number of points in the case of threefolds and the analogous invariants for del Pezzo surfaces.

Keywords: Rational curve; Fano variety; Fano variety of index n − 1; Del Pezzo surface; Lefschetz pencil; normal bundle; Gromov–Witten invariant; enumerative identity

MSC 2010: Primary 14H10; 14N35; Secondary 14N10; 14N25; 14D05

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About the article


Received: 2016-08-08

Published Online: 2018-07-20

Published in Print: 2018-10-25


Funding: While doing this work, I was indirectly supported by the National Science Foundation grant DMS-1308244, “Nonlinear Analysis on Sympletic, Complex Manifolds, General Relativity, and Graphs”.


Citation Information: Advances in Geometry, Volume 18, Issue 4, Pages 451–465, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0010.

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