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On a notion of speciality of linear systems in $ \mathbb{P}^n$

Authors: M. C. Brambilla, O. Dumitrescu and E. Postinghel
Journal: Trans. Amer. Math. Soc. 367 (2015), 5447-5473
MSC (2010): Primary 14C20; Secondary 14J70, 14C17
Published electronically: November 6, 2014
MathSciNet review: 3347179
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Abstract: Given a linear system in $ \mathbb{P}^n$ with assigned multiple general points, we compute the cohomology groups of its strict transforms via the blow-up of its linear base locus. This leads us to give a new definition of expected dimension of a linear system, which takes into account the contribution of the linear base locus, and thus to introduce the notion of linear speciality. We investigate such a notion, giving sufficient conditions for a linear system to be linearly non-special for an arbitrary number of points and necessary conditions for a small number of points.

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Additional Information

M. C. Brambilla
Affiliation: Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, via Brecce Bianche, I-60131 Ancona, Italy

O. Dumitrescu
Affiliation: Department of Mathematics, MSB 2107, University of California, Davis, California 95616

E. Postinghel
Affiliation: Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland

Keywords: Linear systems, fat points, base locus, linear speciality, effective cone
Received by editor(s): October 24, 2012
Received by editor(s) in revised form: June 1, 2013
Published electronically: November 6, 2014
Additional Notes: The first author was partially supported by Italian MIUR funds
The second author is a member of “Simion Stoilow” Institute of Mathematics of the Romanian Academy (
The third author was partially supported by Marie-Curie IT Network SAGA, [FP7/2007-2013] grant agreement PITN-GA-2008-214584
All authors were partially supported by Institut Mittag-Leffler.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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