Variation of Ramification Loci of Generic Projections

✍ Scribed by Hubert Flenner; Mirella Manaresi

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 645 KB
• Volume: 194
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19981940107

✦ Synopsis

A b s t r a c t . Let. A' P N = P i be a subvariety of dimension n and A P N be a generic linear subspace of dimension Nk -1 with k 2 n. Then the linear projection TI\ : X -+ P' is a finite map. Let R(x,j) be its ramification locus. In this paper we study the map from the Grassmannian G ( Nk -1, N ) of planes of dimension Nk -I in P N to the Hilbert moduli space given by A I+ R(x,j). We wish to compute in particular the dimension, say, 1) of the image of this map. T h e motivation of this question comes from the fact that these ramification cycles are closely related t o the Stiickrad -Vogel cycle. We show that 7) is just the transcendence degree of a certain part of this cycle. The main result is that, under some mild hypothesis, i n case of a projection A' --t P", i.e., k = n , the map A H R ( T , ~) is generically finite and so 1) takes its maximal possible value. Moreover, we show that in the caSe of smooth surfaces A' C P4 and generic projections onto P3 this map is again generically finite if the normal bundle of A ' in P4 is sufficiently positive.

VOGEL constructed a cycle t~ = v(X, Y) = C vi where vi is a cycle of dimension i