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

Pizzato, Marco; Sano, Taro; Tasin, Luca

Algebra Number Theory 11, No. 10, 2369-2395 (2017).

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).

Reviewer: Roberto Munoz (Madrid)

MSC:

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

Keywords:

weighted complete intersections; nonvanishing; Ambro-Kawamata conjecture

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