Open subset with non-acyclic structure sheaf

Question

Let $X$ be a projective algebraic variety with acyclic structure sheaf, can we find an open set $U\subset X$, which is the complement of a divisor, such that $\mathcal{O}_U$ is not an acyclic sheaf?

Answer 1

I guess this doesn't quite answer your question, since it sounds like you want to always be able to find such an open set. It might be of interest nonetheless.

The following is a "cheating" example, for two reasons:

• $X \smallsetminus U$ is not irreducible; and
• $X \smallsetminus U$ is forced to be a divisor by a blowup.

Consider $\mathbf{P}^2$ with coordinates $x,y,z$, let $$X = \operatorname{Bl}_{[0:0:1]}\mathbf{P}^2.$$ This is projective with acyclic structure sheaf by [Hartshorne, Ch. V, Prop. 3.4]. Now consider the divisor $D = E + \ell$, where $E$ is the exceptional divisor and $\ell$ is the strict transform of a line $\{z = 0\}$. Then, $$U = X \smallsetminus D \simeq \mathbf{A}^2 \smallsetminus \{0\}$$ is such that $\mathcal{O}_U$ is not acyclic by [Hartshorne, Ch. III, Exc. 4.3].

More generally, let $Z$ be an affine variety and $Y$ a closed subvariety (not necessarily irreducible) of codimension $>1$. An answer by David Speyer shows that $U = Z \smallsetminus Y$ has non-acyclic structure sheaf. If $Z$ has a projective compactification $\widetilde{Z}$ with acyclic structure sheaf, then the complement $\widetilde{Z} \smallsetminus Z$ can be turned into a divisor by a blowup without changing the acyclicity of the structure sheaf by [Chatzistamatiou–Rülling, Thm. 1.1], giving the example you want.

I do not know an example where the two "cheating" tricks are not used.