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A vanishing theorem and symbolic powers of planar point ideals

Part of: (Co)homology theory Curves Surfaces and higher-dimensional varieties Cycles and subschemes General commutative ring theory Arithmetic rings and other special rings

Published online by Cambridge University Press:  01 October 2013

M. Dumnicki ,
T. Szemberg  and
H. Tutaj-Gasińska
M. Dumnicki
Affiliation:
Institute of Mathematics,Jagiellonian University,Łojasiewicza 6, PL-30-348 Kraków,Poland email [email protected]@im.uj.edu.pl
T. Szemberg
Affiliation:
Institute of Mathematics,Pedagogical University of Cracow,Podchorazych 2, PL-30-084 Cracow,Poland email [email protected]
H. Tutaj-Gasińska
Affiliation:
Institute of Mathematics,Jagiellonian University,Łojasiewicza 6, PL-30-348 Kraków,Poland email [email protected]@im.uj.edu.pl

Abstract

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The purpose of this paper is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We then apply this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.

MSC classification

Secondary: 14C20: Divisors, linear systems, invertible sheaves 14F17: Vanishing theorems 14H50: Plane and space curves 14J26: Rational and ruled surfaces 13A15: Ideals; multiplicative ideal theory 13F20: Polynomial rings and ideals; rings of integer-valued polynomials
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 16 , October 2013 , pp. 373 - 387
Copyright
© The Author(s) 2013 

References

Ciliberto, C., ‘Geometric aspects of polynomial interpolation in more variables and of Waring’s problem’, European Congress of Mathematics, Vol. I (Barcelona, 2000), Progress in Mathematics 201 (Birkhäuser, Basel, 2001) 289–316.CrossRefGoogle Scholar
Dumnicki, M., ‘Symbolic powers of ideals of generic points in ${ \mathbb{P} }^{3} $ ’, J. Pure Appl. Algebra 216 (2012) 1410–1417.Google Scholar
Dumnicki, M. and Jarnicki, W., ‘New effective bounds on the dimension of a linear system in ${ \mathbb{P} }^{2} $ ’, J. Symbolic Comp. 42 (2007) 621–635.Google Scholar
Ein, L., Lazarsfeld, R. and Smith, K., ‘Uniform bounds and symbolic powers on smooth varieties’, Invent. Math. 144 (2001) 241–252.CrossRefGoogle Scholar
Geramita, A., Gimigliano, A. and Pitteloud, Y., ‘Graded Betti numbers of some embedded rational $n$ -folds’, Math. Ann. 301 (1995) 363–380.CrossRefGoogle Scholar
Harbourne, B. and Huneke, C., ‘Are symbolic powers highly evolved?’, J. Ramanujan Math. Soc. 28 (2013) 311–330.Google Scholar
Hochster, M. and Huneke, C., ‘Comparison of symbolic and ordinary powers of ideals’, Invent. Math. 147 (2002) 349–369.CrossRefGoogle Scholar
Sullivant, S., ‘Combinatorial symbolic powers’, J. Algebra 319 (2008) 115–142.CrossRefGoogle Scholar
Xu, G., ‘Curves in ${ \mathbb{P} }^{2} $ and symplectic packings’, Math. Ann. 299 (1994) 609–613.CrossRefGoogle Scholar