Arithmetically Cohen–Macaulay bundles on complete intersection varieties of sufficiently high multidegree

Abstract

Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in the case of threefolds.

Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

“…Although the problem of deciding the splitting of vector bundles is very classical, only relatively few cases have been settled: there are cohomological criteria for products of projective spaces and quadrics [10,4], Grassmannians [25,23], hypersurfaces in projective spaces [27,5]. Splitting criteria corresponding to restrictions are useful because they yield dimensional reductions: the problem is reduced to a (usually much) lower dimensional variety, where one can use further cohomological tools.…”

Remarks on higher-rank ACM bundles on hypersurfaces

“…The first higher rank instance of the above statements, namely that of rank 2 ACM bundles on hypersurfaces in P 4 is well understood (see, for instance, [1,2,8,9,10,12]). The most general splitting results known so far are: Date: October 26, 2018.…”

“…2 of degree d. Since E is normalized, χ(E(−1)) 0. Applying the Riemann-Roch theorem to the bundle E(−1) gives usχ(E(−1)) = deg(E(−1)) + r(1 − g) 0,where g is the genus of C. Since C is planar, we have g = (d − 1)(d − 2)/2.…”

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

“…The study of ACM vector bundles of rank 2 on hypersurfaces has received significant attention in the recent past (see [1,[6][7][8]17,[20][21][22]25,3]). …”