Abstract
AbstractIn this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$ . In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.
Highlights
Let X be a Cohen–Macaulay scheme of pure dimension and G a coherent sheaf on X of rank r and homological dimension ≤ 1 – that is, locally over X, there is a two-step resolution 0 → F → E → G → 0, where F and E are finite locally free sheaves. (If X is regular, this condition on G is equivalent to ExtiX (G,OX ) = 0 for all i ≥ 2.) The projectivization π : P(G ) := ProjX SymOX G → X of G is generically a projective bundle with fiber Pr−1; the dimension of the fiber of π jumps along the degeneracy loci of G
The following applications parallel the applications of the projectivization formula in the study of derived categories [25]
The main theorem of this paper implies the corresponding Chow-theoretic version of the formula: for any k ≥ 0, there is an isomorphism of integral Chow groups
Summary
This paper provides two approaches to proving our theorem, one under each of the conditions (A) and (B) The idea behind both approaches is that one could view the projectivization phenomenon as a combination of the Cayley trick and flips. These results are of independent interest on their own. Theorem 3.1 and Corollary 3.4 that the Chow group (and., motive) of every complete intersection variety can be split-embedded into that of a Fano variety (see Example 3.5; compare [28]). The first examples of the theorem are universal Hom spaces (see §4.3.1), flops from Springer-type resolutions of determinantal hypersurfaces (see §4.3.2), and a blowup formula for blowing up along Cohen–Macaulay codimension 2 subschemes (see §4.3.3)
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