Recent zbMATH articles in MSC 14K99https://zbmath.org/atom/cc/14K992024-03-13T18:33:02.981707ZWerkzeugHigher syzygies on general polarized abelian varieties of type \((1, \dots, 1, d)\)https://zbmath.org/1528.140102024-03-13T18:33:02.981707Z"Ito, Atsushi"https://zbmath.org/authors/?q=ai:ito.atsushi-mLet \((X, L)\) be a general polarized abelian variety of type \((d_1, \ldots , d_g)\). It is notoriously difficult to study the projective normality and higher syzygies of \((X, L)\) when \(d_1 = 1\), that is, if \(L\) is a primitive line bundle. In the present paper, the author considers the case \((d_1, \ldots , d_g) = (1, \ldots , 1, d)\) proving that, for a general polarized abelian variety of this type, \(L\) defines a projectively normal embedding if \(d \geq 2^{g+1} - 1\), where \(g = \dim X\). Morever, \(L\) satisfies property \((N_p)\) if
\[
d \geq \frac{(p+2)^{g+1} - 1}{p+1},
\]
where \(p \geq 0\) is an integer. The property \((N_p)\) means, roughly speaking, that \(L\) is projectively normal and the first \(p\) syzygies of the embedding given by \(L\) are ``linear''. The \(p=0\) case solves a conjecture of \textit{L. F. García} in [Arch. Math. 85, No. 5, 409--418 (2005; Zbl 1082.14046)]. It should be noted that this case is particularly interesting, as the author's theorem is optimal. Indeed, the condition \(d \geq 2^{g+1} - 1\) is necessary to get the projective normality of \(L\).
The method of proof consists in giving estimates (see Theorem 1.5) of the basepoint-freeness threshold of \((X, L)\), which in turn implies the result on syzygies by Corollary E of [\textit{Z. Jiang} and \textit{G. Pareschi}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 4, 815--846 (2020; Zbl 1459.14006)] and Theorem 1.1 of the reviewer [Algebra Number Theory 14, No. 4, 947--960 (2020; Zbl 1442.14140)]. Due to the upper-semicontinuity of the basepoint-freeness threshold, these estimates are obtained by some explicit calculations on a suitable polarization on a product of elliptic curves.
Finally, we point out that a result about the projective normality of general polarized abelian varieties of any type has been recently proved by the author in [Bull. Lond. Math. Soc. 55, No. 6, 2793--2816 (2023; Zbl 07779847)].
Reviewer: Federico Caucci (Ferrara)