We show that the Ambro–Kawamata nonvanishing conjecture holds true for a quasismooth WCI which is Fano or Calabi–Yau, i.e., we prove that, if is an ample Cartier divisor on , then is not empty. If is smooth, we further show that the general element of 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.
"Effective nonvanishing for Fano weighted complete intersections." Algebra Number Theory 11 (10) 2369 - 2395, 2017. https://doi.org/10.2140/ant.2017.11.2369