© Publications mathematiques de l’I.H.E.S., 1969, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.

/pdf/the-irreducibility-of-the-space-of-curves-of-given-genus-2llw91qb4n.pdf

The irreducibility of the space of curves of given genus

© Publications mathématiques de l’I.H.É.S., 1994, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/moduli-of-representations-of-the-fundamental-group-of-a-2tho5syb3n.pdf

Moduli of representations of the fundamental group of a smooth projective variety I

IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian category of such representations is denoted by mod-A.) Our motivation is twofold. Firstly, such moduli spaces should play an important role in organising the representation theory of wild algebras. Secondly, such moduli spaces can be identified with moduli spaces of vector bundles on special projective varieties. This identification is somewhat hidden in earner work ([6], [7]) but has become more explicit recently ([4], [12]). It can now be seen to arise from a 'tilting equivalence' between the derived category of mod-A and the derived category of coherent sheaves on the variety. It is well-established that mod-A is equivalent to the category of representations of an arrow diagram, or 'quiver', Q by linear maps satisfying certain 'admissible' relations. Thus, the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classifying orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a) of the representation space 9t{Q, a) of the quiver. Now, the moduli spaces provided by classical invariant theory ([1], [18]) are not interesting in this context. This is because the classical theory only picks out the closed GL(a)-orbits in VA{a), which correspond to semisimple /4-modules, and the quiver Q is chosen so that there is only one semisimple A -module of each dimension vector. On the other hand, we can apply Mumford's geometric invariant theory, with the trivial linearisation twisted by a character x of GL(a), which restricts our attention to an open subset of VA(a), consisting of semistable representations. Within this open set there are more closed orbits and the corresponding algebraic quotient is then a more interesting moduli space. In fact, this approach also has a classical flavour, since it involves the relative (or semi-) invariants of the GL(a) action. The main purpose of this paper is to show that the notions of semistability and stability, that arise from the geometric invariant theory, coincide with more algebraic notions, expressed in the language of mod-A Indeed, the definition is formulated for an arbitrary abelian category as follows:

/pdf/moduli-of-representations-of-finite-dimensional-algebras-3gn716198s.pdf

Moduli of representations of finite dimensional algebras

© Publications mathématiques de l’I.H.É.S., 1999, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/a-1-homotopy-theory-of-schemes-4p2rnq76u8.pdf

$A^1$-homotopy theory of schemes

/pdf/the-geometry-and-cohomology-of-some-simple-shimura-varieties-4tssmg8muy.pdf

The Geometry and Cohomology of Some Simple Shimura Varieties.

In 1934, DuVal [3] listed the configurations of curves which can be obtained by resolving certain isolated double points of embedded surfaces (they are depicted in the figure below). These configurations arise naturally in other contexts, for instance as exceptional curves for pluricanonical embeddings of surfaces [7], and so it seems desirable to have a converse result, showing that a singularity giving rise to such a configuration is necessarily a double point. We have reconsidered the question in a more general context, and obtain in addition the correct nunferical characterization of "rational" singularities (cf. definition below). This characterization is made without the assumption that the surface is embedded and points out the connection of Du Val's work with Castelnuovo's criterion for exceptional curves [2], ([8], p. 38). Finally, we list the configurations obtained from rational triple points.

http://www.jstor.org/stable/pdfplus/2373050.pdf

On isolated rational singularities of surfaces.

© Publications mathématiques de l’I.H.É.S., 1969, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/algebraic-approximation-of-structures-over-complete-local-522ugu63uv.pdf

Algebraic approximation of structures over complete local rings

Graded algebras of global dimension 3

Noncommutative Projective Schemes

On finite simple groups of characteristic 2 type . On the explicit solvability of certain transcendental equations . Algebraic approximation of structures over complete local rings

Michael Artin

Papers

On isolated rational singularities of surfaces.

Algebraic approximation of structures over complete local rings

Graded algebras of global dimension 3

Noncommutative Projective Schemes

On finite simple groups of characteristic 2 type . On the explicit solvability of certain transcendental equations . Algebraic approximation of structures over complete local rings