Arithmetically Cohen-Macaulay Bundles on complete intersection varieties of sufficiently high multidegree

TL;DR

This paper extends the splitting results of rank-two arithmetically Cohen-Macaulay bundles from hypersurfaces to complete intersection varieties of high multidegree, showing they decompose into line bundles in higher dimensions.

Contribution

It generalizes known splitting theorems for ACM bundles from hypersurfaces to complete intersections with high multidegree, including partial results for threefolds.

Findings

ACM bundles of rank two on high-degree complete intersections split into line bundles

Results hold for varieties of dimension at least four

Partial results obtained for three-dimensional cases

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.