# nonvanishing higher cohomology of a very ample divisor

I am looking for smooth projective varieties $$X$$, with $$h^i(X, \mathcal{O}_X) = 0$$ for $$i > 0$$, with a very ample line bundle $$L$$ with some nonvanishing higher cohomology.

What is clear:

(1) Curves will not work (because $$\mathbf{P}^1$$ is the only such curve)

(2) Fano varieties will not work, by Kodaira vanishing (in char. 0)

(3) Hypersurfaces and complete intersections will not work, because the only ones with $$h^i(X, \mathcal{O}_X) = 0$$ for $$i > 0$$ will be Fano

Maybe $$X$$ can be a surface of non-negative Kodaira dimension with $$p_g = q = 0$$, e.g. general type or Enriques?

(Note that the question is easy if I just wanted ample $$L$$, not very ample, as $$L = \mathcal{O}(K_X)$$ on a Godeaux surface or a fake projective plane would be such an example.)

• Take a hypersurface in $\mathbb{P}^2\times\mathbb{P}^1$ of bidegree $(d,1)$ for $d>3$ with the very ample invertible sheaf of bidegree $(1,1)$. – Jason Starr Oct 19 at 0:51
• Thanks, Jason, that's great! I see that the projection of $X$ to the the first factor $\mathbf{P}^2$ identifies $X$ with a blow up of $\mathbf{P}^2$ in $d^2$ points obtained from a pair of intersecting degree $d$ plane curves, in particular $X$ is rational and so $p_g(X) = q(X) = 0$. Furthermore, I see how for large $d$, $\chi(L) < 0$, hence $L$ has nonvanishing $h^1$. – Evgeny Shinder Oct 19 at 11:12
• You are welcome. – Jason Starr Oct 19 at 20:21