N. Mohan Kumar

Arithmetically Cohen–Macaulay bundles on hypersurfaces

We prove that any rank two arithmetically Cohen?Macaulay vector bundle on a general hypersurface of degree at least three in P5 must be split.

Spaces of rational curves on complete intersections

We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree (d1,...., dm) in Pn is irreducible of the expected dimension if Σi=1m di < 2n/3 and n is large enough. This generalizes the results of Harris, Roth and Starr, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if Σi=1m di is small compared to n.

Construction of low rank vector bundles on ⁴ and ⁵

We describe a technique which permits a uniform construction of a number of low rank bundles, both known and new. In characteristic two, we obtain rank two bundles on P 5 . In characteristic p, we obtain rank two bundles on P 4 and rank three bundles on P 5 . In arbitrary characteristic, we obtain rank three bundles on P 4 and rank two bundles on the quadric S 5 in P 6 .

Vector Bundles on Projective Spaces

We generalise a criterion of G. Kempf for a vector bundle on projective space to be decomposable. We also give a similar criterion for a vector bundle to be homogeneous.

Buchsbaum bundles on Pn

Abstract It is shown that there is no indecomposable rank two bundle on P C n , n ≥4, whose first cohomology module is 1-Buchsbaum. It is also shown that there is no indecomposable rank two bundle on P C n , n ≥5, whose first cohomology module is 2-Buchsbaum.

ACM bundles, quintic threefolds and counting problems

We review some facts about rank two arithmetically Cohen-Macaulay bundles on quintic threefolds. In particular, we separate them into seventeen natural classes, only fourteen of which can appear on a general quintic. We discuss some enumerative problems arising from these.

Hilbert scheme components in characteristic 2

We study the deformations of restrictions to P 3 and P 4 of Tangos rank 2 bundle on P 5 (which exists in characteristic 2). Using this, we construct an example of a family of rank two bundles on P 3 (in characteristic 2) with changing α-invariant and an example of a component of the Hilbert scheme of smooth surfaces in P 4 which exists in characteristic 2 but not in anv other characteristic.

Remarks on Unimodular Rows

If (a,b,c) is a unimodular row over a commutative ring A and if the polynomial \({z}^{2} + bz + ac\) has a root in A, we show that the unimodular row is completable. In particular, if 1∕2∈A and b 2−4ac has a square root in A, then (a,b,c) is completable.

Degenerating families of rank two bundles

We construct families of rank two bundles e t on P 4 , in characteristic two, where for t ¬= 0, e t is a sum of line bundles, and e 0 is non-split. We construct families of rank two bundles e t on P 3 , in characteristic p, where for t ¬= 0, e t is a sum of line bundles, and e 0 is non-split.