Algebraic Geometry and Singularities by Vincent Cossart (auth.), Antonio Campillo López, Luis

By Vincent Cossart (auth.), Antonio Campillo López, Luis Narváez Macarro (eds.)

The concentration of this quantity lies on singularity concept in algebraic geometry. It contains papers documenting contemporary and unique advancements and techniques in topics corresponding to solution of singularities, D-module idea, singularities of maps and geometry of curves. The papers originate from the 3rd overseas convention on Algebraic Geometry held in l. a. Rábida, Spain, in December 1991. due to the fact then, the articles have passed through a meticulous technique of refereeing and development, and so they were prepared right into a finished account of the cutting-edge during this field.

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On trouve dans [91 une reponse positive partielle a une question qui en derive. On suppose ici que (V, 0) est normaL Les composantes irreductibles E 1 , ... , Es de E = 7r- 1 (0) sont alors des courbes projectives. : appartenant a un ouvert de Zariski de Ei precise dans chaque enonce. La condition de transversalite exigee ici est algebrique. Elle signifie que 8 = h(o) est un po~t non singulier de Ei et que si Pest l'ideal de 0 definissant E i , ord t P 0 h = 1. x,a Probleme 3-3. Soit (V, 0) une singularite de surface normale et so it 0) un germe de morphisme formeL On suppose qu'il existe une courbe non singuliere h: (T,o) ---- (k 2, 0) te11e que la courbe yoh: (T, 0) -; (V, 0) soit "generique" au sens precedent.

O est k-isomorphe a l'anneau de series formelles k[t], on dira plutot que r E C. Si C =1= 0, on dira que 8 possede la propric~te (el) en O. Les resultats enonces dans ce section sont demontres dans [61. Definition 5-1. Soit (8,0) une singularite d'une surface reduite, 6quidimensionnellc et soit 1T: X -+ (8,0) un morphisme propre et birationnel. TImal de Os,o. On a Zx = 0 si et seulement si 1T est un morphisme fini. E/mOX,E' On remarque aussi que si 1T est une desingularisation de (8,0), Zx est Ie cycle defini par la partie divisorielle de max ; enfin, si Ie morphisme 1T se factorise par l'eclatement de 8 de centre 0, Zx provient d'un diviseur de Cartier sur X et pour toute courbe E C 1T- 1 (0), Ie Hombre d'intersection (Zx .

Et Lejeune-Jalabert j\tl. Modeles canoniques plonges. Kodai Math. J. 14 n 2 1991 ppI94-209. [6] Gonzalez-Sprinberg G. et Lejeune-Jalabert M. COJ,1r~es lisses, cycle maximal et points infiniment· voisins des singularites de surface. prepub196 1992 , a paraitre. [7] Gonzalez-Sprinberg G. Desingularisation des surfaces par des modifications de Nash normalisees. vL Fonction de Artin et germes de courbes tracees sur un germe d'espace analytique a paraitre Amer. J. -1ath. [9] Lejeune-Jalabert M. Arcs analytiques et resolution minimale des singularites des surfaces quasi-homogenes.

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