![]() Harris, J., Morrison, I.: Moduli of Curves. Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif II, Ann. In: Extraits du Séminaire Bourbaki, 1957–1962. Grothendieck, A.: Fondements de la géométrie algébrique: Technique de descente et théorèmes d’existence en géométrie algébrique IV, Les schémas de Hilbert (t. Goto, S., Watanabe, K.: On graded rings, I. Gieseker, D.: On the moduli of vector bundles on an algebraic surface. 90, 1–15 (1996)įulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. Algebra 25, 1413–1441 (1997)įantechi, B., Pardini, R.: On the Hilbert scheme of curves in higher-dimensional projective space. 3, 799–807 (1992)įantechi, B., Pardini, R.: Automorphisms and moduli spaces of varieties with ample canonical class via deformations of abelian covers. New York: Dekker 1999Įllia, P., Hirschowitz, A., Mezzetti, E.: On the number of irreducible components of the Hilbert scheme of smooth space curves. In: Commutative algebra and algebraic geometry (Ferrara). 27, 433–446 (1987)Įllia, P., Hartshorne, R.: Smooth specializations of space curves: Questions and examples. New York: Springer 1995Įllia, P.: D’autres composantes non reduites de Hilbℙ 3. ![]() 8, 423–431 (1975)Įisenbud, D.: Commutative Algebra with a View toward Algebraic Geometry. Preprint 2005, math.AG/0511455, submitted for publicationĮllingsrud, G.: Sur le schéma de Hilbert des variétés de codimension 2 dans ℙ e à cône de Cohen-Macaulay. Algebra 201, 49–61 (2005)Įaston, R.: Surfaces violating Bogomolov-Miyaoka-Yau in positive characteristic. Preprint 2004, math.AG/0405299v1Ĭutkosky, S.D., Ha, H.T.: Arithmetic Macaulayfication of projective schemes. Wajnryb, B.: Diffeomorphism of simply connected algebraic surfaces. Berlin: Springer 1988Ĭatanese, F.: Everywhere nonreduced moduli space. In: Theory of moduli (Montecatini Terme, 1985). 26, 67–88 (1974)Ĭatanese, F.: On the moduli spaces of surfaces of general type. 27, 165–189 (1974)īurns Jr., D.M., Wahl, J.M.: Local contributions to global deformations of surfaces. The essential starting point is Mnëv’s universality theorem.Īrtin, M.: Versal deformations and algebraic stacks. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. It is usually difficult to compute deformation spaces directly from obstruction theories. ![]() (Of course the results hold in the holomorphic category as well.) Similarly one can give a surface over \(\mathbb\) that lifts to ℤ/ p 7 but not ℤ/ p 8. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |