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Clusters of infinitely near points

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Clusters of infinitely near points Antonio Campillo1, Gerard Gonzalez-Sprinberg2, Monique Lejeune-Jalabert2 1 Departamento de Algebra y Geometrıa, Universidad de Valladolid, E-47005 Valladolid (Spain), email: . 2 Institut Fourier, Universite de Grenoble I, UMR 5582U, BP74, 38402 Saint Martin d'Heres (France), email: , Summary. In this article we further develop Zariski-Lipman theory of finitely supported complete (f.s.c.) ideals from the geometric point of view of clusters of infinitely near points. The main results are: a combinatorial characterization of the proximity relation as a generalized Enriques diagram; an explicit description of the monoid of f.s.c. monomial ideals as a polhyedral cone; a construction of such ideals by means of Newton polytopes; and the existence of a natural embedded resolution of a general complete intersection singularity associated to a f.s.c. ideal. Introduction In [E.C] (L. IV, chap. II, 17), Enriques and Chisini consider systems of plane curves which pass with assigned multiplicities through an assigned set of infinitely near points of a (proper) point of the plane.

  • point blowing-ups

  • reducible curves

  • qi ≥

  • irreducible curves contracted

  • zariski's theory

  • linear equivalence

  • ideals endowed

  • ideals

  • any point


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Clusters of infinitely near points
Antonio Campillo1bergprin,Gare´oGdrlaznS-ze2, Monique Lejeune-Jalabert2 1Departamento de Algebra y Geometr´ıa, Universidad de Valladolid, E-47005 Valladolid (Spain), email: campillo@cpd.uva.es. 2Institut Fourier, Universit´e de Grenoble I, UMR 5582U, BP74, 38402 Saint Martin d’H`eres (France), email: gonsprin@fourier.ujf-grenoble.fr, lejeune@fourier.ujf-grenoble.fr
Summarythis article we further develop Zariski-Lipman theory of finitely supported complete (f.s.c.)  ideals. In from the geometric point of view of clusters of infinitely near points. The main results are: a combinatorial characterization of the proximity relation as a generalized Enriques diagram; an explicit description of the monoid of f.s.c. monomial ideals as a polhyedral cone; a construction of such ideals by means of Newton polytopes; and the existence of a natural embedded resolution of a general complete intersection singularity associated to a f.s.c. ideal.
Introduction
In [E.C] (L. IV, chap. II,§17), Enriques and Chisini consider systems of plane curves which pass with assigned multiplicities through an assigned set of infinitely near points of a (proper) point of the plane. They prove that there exist curves with effective multiplicities equal to the virtual ones if and only if some inequalities on the virtual multiplicities, the now so-called “proximity inequalities” hold (at least if no condition is imposed on the degree of the curve).
Later on, Zariski introduces the notion of a complete ideal to give an ideal-theoretic treatment of the previous geometric theory ([Z2], [Z.S]). One of the main results in Zariski’s theory is that any complete ideal in a two dimensional regular local ring has a unique factorization into simple complete ideals. The set of exponents which appear in the factorization is immediately seen on the proximity inequalities for the linear system corresponding to the ideal.
Almost simultaneously, Du Val extends Castelnuovo’s contractibility criterion to reducible curves on a non singular surface. The matrix of the basis change between two natural basis of the free group generated by the irreducible curves contracted to a point of a non singular surface by a morphism, composition of point blowing-ups, is central in the direct analysis. By applying this matrix to a system of virtual multiplicities assigned to these points, one gets the first member of the proximity inequalities.
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These results do not extend directly to higher dimensional case and actually no substantial progress was achieved during fifty years. It is only recently that, by allowing factors with negative exponents, J. Lipman was able to recover a unique factorization statement. The result holds for finitely supported complete (f.s.c.) ideals in a regular local ring of any dimension. This condition means that the ideal is supported at the closed point and that there exists a finite succession of point blowing-ups which makes it invertible. As in dimension two, the special-simple ideals admitted as factors are in one to one correspondence with finite chains of infinitely near points.
The object of this paper is to further investigate Zariski-Lipman’s theory in higher dimension, first by developing a geometric approach of f.s.c. ideals in which the three points of view outlined above for the case of dimension two join together, next by giving explicit description of the monoid of monomial f.s.c. ideals endowed with theand finally, by starting the study of singularities-product naturally arising from f.s.c. ideals.
In section 1, after fixing the language of constellations of infinitely near points, clusters and prox-imity relations, a dictionary between f.s.c. ideals, idealistic clusters and some exceptional divisors on the blown-up variety (or sky) of the constellation is easily got. A synthetic presentation of Zariski and Lipman’s factorization theorem (cor. 1.28 and th. 1.29) is derived from this dictionary through prop. 1.11 and 1.25. A characterization of the proximity relation on an abstract tree is also given (th. 1.6).
Section 2 is devoted to the toric case in any dimension. It turns out that the monoid of idealistic clusters with a given constellation is isomorphic to the monoid of exceptional divisors in the dual of the coneN E1-cycles on the sky of the constellation. Thisof effective exceptional  isomorphism holds also for general clusters (not necessarily toric) in dimension two. More precisely, the main results are the following : a combinatorial description of toric constellations (prop. 2.1 and 2.2), the characterization of toric idealistic clusters in terms of linear proximity inequalities, an algorithm for obtaining the Newton polytope of the monomial f.s.c. ideal corresponding to an idealistic cluster (th. 2.10 to rem. 2.13), an explicit description of the special-simple ideals (lemma 2.16), formulae for the exponents in the-factorization of a monomial f.s.c. in terms of linear ideal proximity relations (th. 2.18), conditions for the non negativity of the exponents (cor. 2.20), the determination of the extremal vectors of the coneN E(th. 2.22 and cor. 2.23).
In section 3, the sky of a constellation is shown to be a natural embedded resolution of complete intersections defined by general elements in a f.s.c. ideal with this constellation, provided the characteristic of the ground field is zero. In this set-up, the cluster replaces the Newton polytope of [Kh] and [V].
The references given at the end do not pretend to be exhaustive; some of them are related to recent works on other aspects or other points of view on this subject.
Notations. —Throughout this paper, a variety will mean an algebraic variety,i.e.a reduced and irreducible scheme of finite type over an algebraically closed fieldK point will mean a. A closed point.
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