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A remark on higher syzygies on abelian surfaces. (English) Zbl 1411.14016

Recall that property \((N_{p})\) for an ample line bundle \(L\) on a projective variety \(X\) roughly consists of a sequence of increasing positivity conditions. To be more precise, property \((N_{0})\) means that the complete linear series determined by \(M\) embeds \(X\) as a projectively normal subvariety. Property \((N_{1})\) means that the homogeneous ideal \(I_{L}\) of the embedding is generated in degree 2. For \(p \geq 2\), property \((N_{p})\) means that the first \(p\) syzygy modules in the projective resolution of \(I_{L}\) are generated by linear terms (for more details, see Chapter 1.8.D of [R. Lazarsfeld, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. Berlin: Springer (2004; Zbl 1093.14501)]).
Given a non-negative integer \(p\) and an abelian surface \(X\) with an ample line bundle \(L\), A. Küronya and V. Lozovanu [“A Reider-type theorem for higher syzygies on abelian surfaces”, Preprint, arXiv:1509.08621] showed that if \((L^{2}) \geq 5 (p+2)^{2}\), then the following are equivalent:
(1) \(X\) does not contain an elliptic curve \(C\) with \((L \cdot C) \leq p+2\);
(2) Property \((N_{p})\) holds for \(L\).
The present paper shows that the above theorem holds with the weaker assumption \((L^{2}) > 4 (p+2)^{2}\).
The idea is to apply a theorem of R. Lazarsfeld et al. [Algebra Number Theory 5, No. 2, 185–196 (2011; Zbl 1239.14035)], which gives sufficient conditions for an ample line bundle on an abelian variety to satisfy property \((N_{p})\), and involves the existence of an effective \(\mathbb Q\)-divisor \(F_{0}\) satisfying certain positivity conditions. To construct \(F_{0}\), the author uses tools “developed in the study of Fujita’s base-point-freeness conjecture” (second page of the present paper).

MSC:

14C20 Divisors, linear systems, invertible sheaves
13D02 Syzygies, resolutions, complexes and commutative rings
14K05 Algebraic theory of abelian varieties
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References:

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