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 conception in algebraic geometry. It contains papers documenting contemporary and unique advancements and strategies in matters comparable to answer of singularities, D-module thought, singularities of maps and geometry of curves. The papers originate from the 3rd foreign convention on Algebraic Geometry held in los angeles Rábida, Spain, in December 1991. considering then, the articles have gone through a meticulous means of refereeing and development, and so they were equipped right into a entire account of the state-of-the-art during this field.

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Over the arcs (Xij, Xej) E G t IXe x Gt I x~ and (xcj,Xj). , Xt;_l E B~ and Xt; E Be - B~. Also the arcs (Xhk, Xt) are redundant for j > i and the arcs (Xhtj' Xhk) are redundant for j < i -1. Thus, one has, on the one hand, the arc (Xhk, Xt;) and, on the other, the path Xht2 ... Xht,_l connected to the product graph G t IXe x G t Ix~ by the arc (Xht;_" Xhk) and to the path Xt2 ... Xti_l by the arcs (Xhtj,Xt,), 2 ~ j ~ i - l . On the contrary, the transitivity fails on G t IXe x G t Ix~, on the connection of this with G t I x~ and on the connection, between them, of the paths Xht2 ...

0 7rl, and such that the strict transform XN of X at ZN is smooth and transversal (normal crossing) to the exceptional divisor EN. The embedded resolution of singularities was extended to analytical spaces in 1974 in [IJ. 1) for each variety X, This leads us to consider an algorithmic point of view (see [5]). Looking for such an algorithmic viewpoint, our initial motivation, in this paper, is to try to understand the behaviour, after blowing up 7ri+l, of the set (or a subset) of subvarieties Ei created at Zi by the sequence of transformations, More precisely, we are interested in the following question: if 7r: Z' --+ Z is a blowing up of a smooth variety at a regular center C, and if C is a smooth geometrical configuration of subvarieties of Z, try to describe the transform C' of C in Z', i,e" the set of strict transforms of subvarieties in C not included in C and the fibers of those contained in C, as well as the irreducible components of the intersections of such objects.

07l'r la desingularisation plongee minimale d'une branche plane (V, 0) comme ci-dessus et soit m = (mo, ... , mi,"') la suite des multiplicitCs de Nash de sa normalisation. Si VEe est une branche d6nnie parametriquement par h: (T, 0) ---+ (k 2 ,0) telle que pih) E 'Hmo, ... ,m J , 0 :::; j :::; i (ef Rcmarquc 4-5), pour un en tier i 2: {3g = inf {i I mi = I}, alors 1. pest aussi Ia d6singularisation plongee minimale de V 2. V passe par Ie point de V situe dans Ie (i - ,6g nitesimal de O. + r)-eme Yoisinage infl- Si PHd"h) ~ Tr(i+l),alors V et V n'ont pas de point commun dans un Yoisinage innnitesimal de 0 d 'ordre > i - :3g + r.

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