On a notion of speciality of linear systems in P^n

Given a linear system in 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 arbitrary number of points, and necessary conditions for small numbers of points.

[1]  Elisa Postinghel.  A new proof of the Alexander-Hirschowitz interpolation theorem , 2010, 1003.0323.

[2]  G. Ottaviani,et al.  On the Alexander–Hirschowitz theorem , 2007, math/0701409.

[3]  Karen A. Chandler.  The geometric interpretation of Fröberg–Iarrobino conjectures on infinitesimal neighbourhoods of points in projective space , 2003 .

[4]  M. Brambilla,et al.  ON HYPERSURFACES IN P3 WITH FAT POINTS IN GENERAL POSITION , 2009 .

[5]  Ciro Ciliberto,et al.  Geometric Aspects of Polynomial Interpolation in More Variables and of Waring’s Problem , 2001 .

[6]  Ana-Maria Castravet.  THE COX RING OF M0,6 , 2009 .

[7]  M. Dumnicki.  REGULARITY AND NON-EMPTYNESS OF LINEAR SYSTEMS IN P n , 2008 .

[8]  Jinhyung Park,et al.  Cones of divisors of blow-ups of projective spaces , 2011 .

[9]  J. Tevelev,et al.  Hilbert's 14th problem and Cox rings , 2005, Compositio Mathematica.

[10]  I. Dolgachev,et al.  Weyl groups and Cremona transformations , 1983 .

[11]  V. Danilov.  Cohomology of Algebraic Varieties , 1996 .

[12]  A. Laface,et al.  On a class of special linear systems of $\mathbb{P}^3$ , 2006 .

[13]  Marcin Dumnicki,et al.  An algorithm to bound the regularity and nonemptiness of linear systems in Pn , 2009, J. Symb. Comput..

[14]  Antonio Laface,et al.  Standard Classes on the Blow-Up of ℙ n at Points in Very General Position , 2010, 1004.4010.

[15]  Ana-Maria Castravet.  The Cox ring of \overline{}_{0,6} , 2009 .

[16]  E. Ballico,et al.  POSTULATION OF GENERAL QUINTUPLE FAT POINT SCHEMES IN P 3 , 2008, 0810.1372.

[17]  I. Shafarevich.  Algebraic Geometry II , 1996 .

[18]  D. Anick,et al.  Thin algebras of embedding dimension three , 1986 .

[19]  Exceptional Loci on $\bar M_{0,n}$ and Hypergraph Curves , 2008, 0809.1699.

[20]  Anthony Iarrobino,et al.  Inverse system of a symbolic power III: thin algebras and fat points , 1997, Compositio Mathematica.

[21]  C. Ciliberto,et al.  Variations on Nagata's Conjecture , 2012, 1202.0475.

[22]  Ralf Fröberg,et al.  An inequality for Hilbert series of graded algebras. , 1985 .

[23]  M. Dumnicki.  Regularity and non-emptyness of linear systems in $\mathbb P^n$ , 2008, 0802.0925.

[24]  E. Ballico,et al.  Postulation of general quartuple fat point schemes in P3 , 2008 .