## Effective nonvanishing for Fano weighted complete intersections.(English)Zbl 1390.14144

Let $$\mathbb{P}=\mathbb{P}(a_0,\dots,a_n)$$ be a weighted projective space and $$X=X_{d_1, \dots, d_c} \subset \mathbb{P}$$ a well-formed (codimension in $$X$$ of its intersection with the singular locus of $$\mathbb{P}$$ bigger than $$1$$) quasismooth (singularities of $$X$$ are just those of $$\mathbb{P}$$ on $$X$$) complete intersection. The main result of the paper (see Theorem 1.2) states that if $$X$$ is not a linear cone and Fano or Calabi-Yau then any ample Cartier divisor $$H$$ on $$X$$ is effective, that is $$|H| \neq \emptyset$$. Furthermore, if $$X$$ is smooth, the number of weights equal to one is at least the codimension and the general element of $$|\mathcal{O}_X(1)|$$ is smooth; moreover a description of the base locus of this last linear system is provided (see Remark 5.5). In the particular case $$X$$ being a hypersurface (see Theorem 1.3) it is shown that for $$H$$ ample and Cartier, if $$H-K_X$$ is ample then $$|H|$$ is nonempty (positive partial result on Ambro-Kawamata Conjecture, see Conjecture 1.1). Moreover, also for hypersurfaces, $$X$$ Gorenstein and $$H$$ ample then $$K_X+mH$$ is globally generated when $$m \geq n$$ (Fujita’s freenes conjecture in this particular setup).

### MSC:

 14M10 Complete intersections 11D04 Linear Diophantine equations 14J45 Fano varieties
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### References:

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