By Danielle Dias

ISBN-10: 3540620508

ISBN-13: 9783540620501

The most issues of this booklet are to set up the triple formulation with none hypotheses at the genericity of the morphism, and to improve a conception of whole quadruple issues, that is a primary step in the direction of proving the quadruple element formulation less than much less restrictive hypotheses.

This ebook will be of curiosity to graduate scholars and researchers within the box of algebraic geometry. The reader is predicted to have a few simple wisdom of enumerative algebraic geometry and pointwise Hilbert schemes.

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**Additional info for Configuration spaces over Hilbert schemes and applications**

**Sample text**

C- } 033 a~ = ~H*r. ~ * r . ~*{~}~ *: 9 ( G 1 3 x W) 9( c ~ • w). al a n d r Since on (G13 • W ) the restrictions W'il and 033] are equal, one has : 31 = ~ * ( r { ~ } ' : ) . ~ ' r . (c13 • w ) . al = F{~}*:. 10)) w~ = / 7 / o 7? o ~ X W)) . for i =1, 2, one h a s : ~ i , ( ~ * r - (a13 • w)) = p5,~,~,(~*~*p~*r- (G13 • W ) ) . T h e triple formula 37 i But ~,(G13 x W ) = [H2(V) • W] = 1, since Gla is a graph of a m o r p h i s m from H z ( V ) to V. A p p l y i n g (FP) to ~ yields : WI,(~2*F ' (G13 • W)) ~- p~,~,(~-*p'2*F 91) = p~,p~*r.

Iv) Let r be the birational morphism r H3(V) ~ H~(V) • V i ~ ((vl, v2, d~), v3) if [ = (t, d12, d23, d31, Vl, v2, v3). (v) The imbedding/3 : H3(V) x W '-~ H3(V x W) has been defined above. ~ denotes g • idw, where g is a morphism. 2), one has a commutative diagram : 28 Triple formula P~2 A H3(V x W) A H2(V • A W) A H2(V) x H3(V) X W Fw H2(~-~) • W V x W CO 1 pr2 W f J J 7 D V VxVxW J J Prl j J J pl VxW r Diagram 4 ~Varning : p r l is represented by a dotted line, since it does not make the diagram commute ; however, p r l o j = a.

K < n/2, so that the classes defined below are meaningful. 2. Its image consists of the "horizontal" triplets of V x W. tt, ~) V Let us denote by P~' : H3(---~) x W ~ V the morphism which takes (i, w) to vl, where [ = (t, d12, d23, d31, vl, v2, v3) is in H3(~"V). If F is the graph of f, the diagram H3(-~) ~ H3(V-""~W) ~ H-~(V) x W ~ V follows. D e f i n i t i o n 6 : With the above notation, let (a) M3 = j3*[H3(F)] (b) ~-~3= P "1. M 3 e 9 CH3,,_2,,(H3(V) x W) CH3,,_2,,(V) = CH2k(V). m-'-~is said to be the "triple class".

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