By Danielle Dias

ISBN-10: 3540620508

ISBN-13: 9783540620501

The most issues of this e-book are to set up the triple formulation with none hypotheses at the genericity of the morphism, and to strengthen a concept of whole quadruple issues, that's a primary step in the direction of proving the quadruple element formulation less than much less restrictive hypotheses.

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

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

**Example text**

8 ) x c W = (co~e(V) 1 x cW) . (U x W) . But codim(U x W) = n. Applying (T~I) yields: { ( ~ o ( v ) -1. 0 ) x c W } " : { ~ c ( v ) -1 x c w } ...... 9( 0 • w ) = ~*{~}~. (O x w ) 35 The triple formula (recall t h a t ~ = c(V) -1 x c W - see n o t a t i o n 5). For the other terms of this sum, one has : (od~c(V) - 1 . J~q*F h-l) x c W = (od~c(V) -1 X c W ) . ( ( B q ' F h-l) x W ) But c o d i m ( B q * F h-l) = (n + 1) + (h - 1) = n + h. From (741) again, one h a s : {(0J~c(V) - 1 . B q * F h - l ) x cW} TM = {w~c(V) -1 x cW} ......

32) H3(V-~x W) be the imbedding on the horizontal triplets. One would like to show t h a t it is transverse to ~ = E13 + ~3 + E'. Once again, we study what happens close to the most degenerate case : to = (to, do, do, do, 0, 0, 0) where to is amorphous. Let w 9 W. Consider t~o = / 3 ( ~ , w). At 54 Intermediate computations (0, w), one has a chart of V • W : (x, y, 5", z'), where (x, y, 5") is a chart of V at 0 and z~ is a chart of W at w, the ideal of to being (x 2, xy, y2, Z) and the ideal of do being (x 2, y, z).

23), one has the equality: k m-'~ = m 2 f * f , V - c,,f*f,V - f * f , ck q- 2c~. 24) h=l Using the known expression for m2 (see theorem 2), one can give for ~33 the more familiar expression (cf. Kleiman [KL1], th. 9) : k m"~ = f * f , m2 -- 2ckrn2 + ~ 2hc~-hCk+h h=l To conclude, one has shown the following theorem : T h e o r e m 4 Let f : V ----+ W be an arbitrary morphism of proper, smooth varieties with d i m W = k + d i m V and 0 < k < d i m V /2. Let ~33 E CH2k(V) be the "triple class", direct image of the cycle A A M3 = [g3(r)].

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