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Codimension 3 Arithmetically Gorenstein Subschemes of projective N -space

Robin Hartshorne, Irene Sabadini, Enrico Schlesinger (2008)

Annales de l’institut Fourier

We study the lowest dimensional open case of the question whether every arithmetically Cohen–Macaulay subscheme of N is glicci, that is, whether every zero-scheme in 3 is glicci. We show that a general set of n 56 points in 3 admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in 3 .

Cohomology of the G-Hilbert Scheme for 1/r(1,1,R−1)

Kędzierski, Oskar (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 14E15; Secondary 14C05,14L30.In this note we attempt to generalize a few statements drawn from the 3-dimensional McKay correspondence to the case of a cyclic group not in SL(3, C). We construct a smooth, discrepant resolution of the cyclic, terminal quotient singularity of type 1/r(1,1,r−1), which turns out to be isomorphic to Nakamura’s G-Hilbert scheme. Moreover we explicitly describe tautological bundles and use them to construct a dual basis to...

Compactifications des espaces de configuration dans les schémas de Hilbert

Laurent Evain (2005)

Bulletin de la Société Mathématique de France

Soient F ( X , n ) = X n - Δ le complémentaire de l’union Δ des diagonales dans X n et U un quotient (éventuellement trivial) de F ( X , n ) par un sous-groupe du groupe symétrique 𝔖 n . Ce travail présente des procédés de compactification de  U dans des produits de schémas de Hilbert. Notre démarche généralise et unifie des constructions classiques dues à Schubert-Semple, LeBarz-Keel, Kleiman et Cheah. Une étude géométrique plus détaillée est faite pour les cas n 3 . Cette étude inclut notamment une classification complète, la détermination...

Contraction of excess fibres between the McKay correspondences in dimensions two and three

Samuel Boissière, Alessandra Sarti (2007)

Annales de l’institut Fourier

The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres over the origin reveal similar properties. We construct a morphism between these two resolutions, contracting exactly the excess part of the exceptional fibre. This construction is motivated by the study of some pencils of K3 surfaces appearing as minimal resolutions...

Courbes rationnelles sur les variétés homogènes

Nicolas Perrin (2002)

Annales de l’institut Fourier

Soit X une variété homogène sous un groupe G . Nous étudions les orbites maximales de X sous l’action d’un parabolique de G . Nous les décomposons en fibrations affines et projectives. Cette description permet de montrer que le schéma de Hilbert des courbes rationnelles lisses de classe fixée est non vide et irréductible.

Curves on a double surface.

Scott Nollet, Enrico Schlesinger (2003)

Collectanea Mathematica

Let F be a smooth projective surface contained in a smooth threefold T, and let X be the scheme corresponding to the divisor 2F on T. A locally Cohen-Macaulay curve C included in X gives rise to two effective divisors on F, namely the largest divisor P contained in C intersection F and the curve R residual to C intersection F in C. We show that under suitable hypotheses a general deformation of R and P lifts to a deformation of C on X, and give applications to the study of Hilbert schemes of locally...

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