A Generic Cubic Surface Contains No Involutive Curve

Abstract

Ž. In 1988, J. Bernstein and V. Lunts In¨ent. Math. 94, 223᎐243 proved that a generic surface in ‫ސ‬ 3 of degree 4 or greater would not contain a curve which supported a module over the second Weyl algebra. It is known that quadric surfaces and planes always do. This paper shows that a generic cubic surface does not. ᮊ

The characteristic variety of a generic foliation

“…Still under the assumption that k ≥ 4, Lunts extends the above result to arbitrary n ≥ 2 in [7]. For k = 3 and n ≥ 2 the very same statement has been proved by McCune [8]. All these results, in contrast with Stafford's, do not exhibit explicit examples of non-holonomic A n -modules but instead prove that they are generic in the above sense.…”

On Homogenous Minimal Involutive Varieties

“…This result was later generalized by Lunts [15] to all n 2 and k 4, and by T. McCune [16] to k = 3 and n = 2. It should be pointed out that although these results imply that 'most' polynomials of degree k 3 give rise to minimal involutive hypersurfaces in C 4 , the proofs given in [3], [15] and [16] do not allow one to write down any explicit examples of such polynomials -say, one with rational coefficients, with which one might try a few computations.…”

Families of minimal involutive surfaces in projective space

“…They called an involutive homogeneous variety of A 2n minimal if it does not contain any proper involutive homogeneous subvarieties and proved that if M is a cyclic A n -module and ann gr(A n ) (gr(M )) is a prime ideal such that Ch(M ) ⊂ A 2n is minimal involutive homogeneous, then M is irreducible. Moreover, they showed that a generic homogeneous hypersurface of A 4 of degree greater than 3 is minimal involutive homogeneous; a result that was later generalized by Lunts to A 2n [11] and by T. C. McCune [12] to polynomials of degree 3. As a corollary, we have that if n ≥ 2 is an integer and d is a generic operator of degree at least 3 in A n , then A n /A n d is a nonholonomic irreducible module of dimension 2n − 1 > n ≥ 2.…”