Soit la décomposition canonique de l’espace des arcs passant par une singularité normale de surface. Dans cet article, on propose deux nouvelles conditions qui si elles sont vérifiées permettent de montrer que n’est pas inclus dans . On applique ces conditions pour donner deux nouvelles preuves du problème de Nash pour les singularités sandwich minimales.
Le théorème de Belyi affirme que sur toute courbe algébrique lisse projective et géométriquement connexe, définie sur , il existe une fonction non ramifiée en dehors de . Nous montrons que cette fonction peut être choisie sans automorphismes, c’est-à-dire telle que pour tout automorphisme non trivial de , on ait . Nous en déduisons que si est une extension finie de , toute -classe d’isomorphisme de courbes algébriques lisses projectives géométriquement connexes peut être caractérisée...
Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from Rn to R obtained by composition of polynomial functions and the absolute value. Every semipolynomial can be defined as a straight-line program containing only instructions with the following type: polynomial, absolute value, sup and inf and such a program will be called a semipolynomial expression. It will be proved, using the ordinary real positivstellensatz, a general real positivstellensatz concerning...
Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if K is a subfield...
We show that the natural generalization of a conjecture of Hain and Looijenga to the case of pointed curves holds for all and if and only if the tautological rings of the moduli spaces of curves with rational tails and of stable curves are Gorenstein.
A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line , then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.