Higher order embeddings via the basepoint-freeness threshold (Q6583738)

The notion of \(k\)-jet very ampleness and \(k\)-very ampleness of line bundles have been introduced as possible higher analogues of very ampleness by Beltrametti, Francia and Sommese in [\textit{M. Beltrametti} et al., Duke Math. J. 58, No. 2, 425--439 (1989; Zbl 0702.14010)].\N\NLet \(X\) be a complex projective variety and \(L\) a line bundle on \(X\). \(L\) is said to be \(k\)-jet very ample, \(k\geq0\), if the evaluation map \[H^0(X,L)\to H^0(X,L\otimes\mathcal O_X/\mathcal I_{x_1}\cdots \mathcal I_{x_{k+1}})\] is surjective for any, not necessarily distinct, \(k + 1\) points \(x_1,\dots, x_{k+1}\) in \(X\). In particular, 0-jet very ample means that \(L\) is globally generated, and 1-jet very ample means that \(L\) is very ample. \(L\) is said to be \(k\)-very ample if the evaluation map \[H^0(X,L)\to H^0(X,L\otimes \mathcal O_Z)\] is surjective for 0-dimensional subschemes \(Z\subseteq X\) of length \(k + 1\). When \(k \geq 2\), \(k\)-jet very ampleness is in general stronger than \(k\)-very ampleness.\N\NIn the present paper, the author studies these conditions on abelian varieties in relation to the Jiang-Pareschi basepoint-freeness threshold and on Kummer varieties.\N\NThe basepoint-freeness threshold is an invariant \(\beta(A,l)\) for the polarization \(l\in \text{Pic} A/\text{Pic}^0 A\) of an abelian variety \(A\), introduced in [\textit{Z. Jiang} and \textit{G. Pareschi}, Ann. Sci. Éc. Norm. Supér. (4) 53, No. 4, 815--846 (2020; Zbl 1459.14006)]. In the first part of the present paper it is proved that if \[\beta(A,l) <\frac{1}{k+1},\] then \(L\otimes N\) is \(k\)-jet very ample, for any ample line bundle \(L\) representing the polarization \(l\) and any nef line bundle \(N\) on \(A\). This result partially recovers the well-known result of Bauer-Szemberg [\textit{Th. Bauer} and \textit{T. Szemberg}, Math. Z. 224, No. 3, 449--455 (1997; Zbl 0897.14011)] saying that the tensor power of \(m\) ample line bundles on an abelian variety is \(k\)-jet very ample, when \(m\geq k + 2\).\N\NThe second part of the paper studies a Kummer variety \(K(A)\), that is the quotient of an abelian variety \(A\) by the action of the inverse morphism \(p\mapsto -p\). It is known that an ample line bundle on \(K(A)\) is globally generated, and that a square of it is very ample. This is generalized as follows. If \(L\) is an ample line bundle on \(K(A)\) and \[m >\frac{k+1}{2},\] then \(L^{\otimes m}\otimes N\) is \(k\)-very ample, for any nef line bundle \(N\) on \(K(A)\). This complements a previous result of the author on syzygies of Kummer varieties [\textit{F. Caucci}, Trans. Am. Math. Soc. 377, No. 2, 1357--1370 (2024; Zbl 1536.14039)].