Effective nonvanishing for Fano weighted complete intersections

Abstract

We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI $X$ which is Fano or Calabi–Yau, i.e., we prove that, if $H$ is an ample Cartier divisor on $X$, then $\left|H\right|$ is not empty. If $X$ is smooth, we further show that the general element of $\left|H\right|$ is smooth. We then verify the Ambro–Kawamata conjecture for any quasismooth weighted hypersurface. We also verify Fujita’s freeness conjecture for a Gorenstein quasismooth weighted hypersurface.

For the proofs, we introduce the arithmetic notion of regular pairs and highlight some interesting connections with the Frobenius coin problem.