G. V. Ravindra

The Grothendieck-Lefschetz theorem for normal projective varieties

We prove that for a normal projective variety X in characteristic 0, and a base-point free ample line bundle L on it, the restriction map of divisor class groups Cl(X) → Cl(Y ) is an isomorphism for a general member Y ∈ |L| provided that dimX ≥ 4. This is a generalization of the Grothendieck-Lefschetz theorem, for divisor class groups of singular varieties. We work over k, an algebraically closed field of characteristic 0. Let X be a smooth projective variety over k and Y a smooth complete intersection subvariety of X. The Grothendieck-Lefschetz theorem states that if dimension Y ≥ 3, the Picard groups of X and Y are isomorphic. In this paper, we wish to prove an analogous statement for singular varieties, with the Picard group replaced by the divisor class group. Let X be an irreducible projective variety which is regular in codimension 1 (for example, X may be irreducible and normal). Recall that for such X, the divisor class group Cl(X) is defined as the group of linear equivalence classes of Weil divisors on X (see [10], II, §6). If dimX = d, then Cl(X) coincides with the Chow group CHd−1(X) as defined in Fulton’s book [7]. If Y ⊂ X is an irreducible Cartier divisor, which is also regular in codimension 1, there is a well-defined restriction homomorphism

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.

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

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.

Curves on threefolds and a conjecture of Griffiths-Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface X ⊂ P of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in P. We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

EXTENSIONS OF VECTOR BUNDLES WITH APPLICATION TO NOETHER-LEFSCHETZ THEOREMS

Given a smooth, projective variety Y over an algebraically closed field of characteristic zero, and a smooth, ample hyperplane section X ⊂ Y, we study the question of when a bundle E on X, extends to a bundle on a Zariski open set U ⊂ Y containing X. The main ingredients used are explicit descriptions of various obstruction classes in the deformation theory of bundles, together with Grothendieck–Lefschetz theory. As a consequence, we prove a Noether–Lefschetz theorem for higher rank bundles, which recovers and unifies the Noether–Lefschetz theorems of Joshi and Ravindra–Srinivas.

Torsion points and matrices defining elliptic curves

Let k be an algebraically closed field, char k ≠ 2, 3, and let X ⊂ ℙ2 be an elliptic curve with defining polynomial f. We show that any non-trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Φr of size 3r × 3r with linear polynomial entries such that detΦr = fr. We also show that the identity, thought of as the trivial torsion point of order r, determines up to equivalence, a unique minimal matrix Ψr of size (3r - 2) × (3r - 2) with linear and quadratic polynomial entries such that detΨr = fr.

Zero cycles on certain surfaces in arbitrary characteristic

AbstractLetk be a field of arbitrary characteristic. LetS be a singular surface defined overk with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation

The Noether–Lefschetz theorem for the divisor class group

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