Effective nonvanishing for fano weighted complete intersections

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 |H| is not empty. If X is smooth, we further show that the general element of |H| 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.