In the paper “Varieties of small codimension in projective space” Hartshorne has the following conjecture.
Conjecture 5.1. Let A be a regular local ring of dimension n. Let P be a prime ideal of A such that A/P has an isolated singularity. Let r = dim(A/P) and suppose that r > 1/3(2n – 1). Then A/P is a complete intersection.
We don’t have any positive results on this conjecture (as stated) this except in the case where r = n – 1. Namely, if r = n – 1, then A/P is a hypersurface as a regular local ring is a UFD.
We do have a negative result, namely, the conjecture seems to be wrong for n = 6 and r = 4. Namely, Tango constructed a nonsplit rank 2 vector bundle on P^5 in characteristic 2. A general section of a high twist of this vector bundle will give a codimension 2 smooth subvariety of P^5 which is not a complete intersection. Pulling this back to punctured affine 6-space will give the desired counter example.
This is related to what Hartshorne says just before stating his conjecture, namely that his local version is actually stronger than the original conjecture. If we don’t make the conjecture stronger then in 5.1 we would put the inequality r > 1/3(2n + 1). And Tango’s example no longer gives a counter example. Let’s call this the corrected conjecture.
Let us, as is customary, strengthen the corrected conjecture as follows.
Conjecture E.1. Let A be a regular local ring of dimension n with spectrum S and punctured spectrum U. Let V be a closed subscheme of U which is a local complete intersection, whose closure in S is equidimensional of dimension r. Suppose that r > 1/3(2n + 1). Then there exists a complete intersection Z ⊂ S with V = U ∩ Z scheme theoretically.
Conjecture E.1 again holds for r = n – 1. Work in the projective case suggests that E.1 is not much stronger than the corrected conjecture 5.1.
Lemma 1. Let A be a Noetherian local ring with spectrum S. Let E be a vector bundle on the puncture spectrum U of A. Let Z ⊂ S be a complete intersection of codimension c such that E|Z ∩ U is a trivial vector bundle. If depth(A) > 2 + c, then E is a trivial vector bundle.
Proof. Using induction on c we reduce to the case c = 1. Then Z = Spec(B) where B = A/f for some nonzerodivisor f in A. Hence depth(B) > 2. Thus we have H^1(Z ∩ U, O_Z) = 0. Thus we see that E is trivial on a formal neighbourhood of Z ∩ U in U by a standard deformation argument. But the functor from vector bundles on U to vector bundles on the formal completion of U along Z ∩ U is fully faithful by a Lefschetz type result, see Tag 0EKS, and hence we conclude.
Lemma 2. Let A be a Noetherian local ring with spectrum S. Let E be a vector bundle of rank c on the punctured spectrum U of A. Let Z ⊂ S be a complete intersection such that Z ∩ U is the vanishing scheme of a global section s of E. If depth(A) > 2 + c, then E is a trivial vector bundle.
Proof. Note that E|Z ∩ U is the normal bundle of Z in U. Since Z is a complete intersection we find E|Z ∩ U is a trivial vector bundle. We conclude by Lemma 1.
Lemma 3. Let A be a regular local ring. Let B = A/I be a complete intersection of codimension t. If c < 1/3(dim(B) – 2t – 1) and conjecture E.1 holds, then any vector bundle of rank c on the punctured spectrum of B is trivial.
Proof. Let E be a vector bundle of rank c on the punctured spectrum of B. Choose a finite B module M corresponding to E. Choose a “random” element s of m_B^N M for some N ≫ 0. Denote V inside the punctured spectrum of B the vanishing scheme of s. Observe that dim(V) = dim(B) – c = n – t – c where n is the dimension of A. The inequality n – t – c > 1/3(2n + 1) is equivalent to the inequality of the lemma and hence E.1 tells us that V is the intersection of the punctured spectrum of A with a complete intersection Z in Spec(A). Because we chose N ≫ 0 we find that we may choose the first t generators for the ideal of Z to be the t generators for I. Thus we see that Z is a complete intersection in Spec(B). Then we conclude that E is trivial by Lemma 2.
The lemma tells us that complete intersection rings B should have few interesting low rank vector bundles on their punctured spectra provided E.1 is true. But the appearance of the term -2t in the inequality is annoying. By Grothendieck we know that any invertible module on the punctured spectrum of B is trivial if dim(B) > 3. In other words if 1 < 1/3dim(B) or put another way 1 ≤ 1/3(dim(B) – 1). So let’s ask the following question (where we have dropped the -2t and replaced < by ≤ in the inequality).
Question E.2. Let B be a Noetherian local ring which is a complete intersection. Let E be a vector bundle on the punctured spectrum of B. If rank(E) ≤ 1/3(dim(B) – 1), then is E trivial?
So for example if the rank is 2 the inequality gives 7 ≤ dim(B). If true this would be sharp by what we said above. Anyway, I don’t insist on the exact formula for the inequality; I’m not sure why Hartshorne chose 2/3 as the leading coefficient in his inequality. Really, a much more reasonable question is the following.
Question E.3. Given an integer c does there exists an integer n(c) such that if B is a Noetherian local complete intersection of dimension > n(c), then any E vector bundle of rank c on the punctured spectrum of B is trivial?
For those who prefer projective geometry over local algebra, we ask whether there exist indecomposable rank 2 vector bundles on complete intersection varieties of arbitrarily large dimension… Please let me know if you have interesting examples! Thanks.