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The \(k\)-very ampleness and \(k\)-spannedness on polarized abelian surfaces. (English) Zbl 0957.14009

From the introduction: In general, a \(k\)-very ample line bundle \(L\) gives naturally a notion of \(k\)-th order embedding as follows. Let \(S^{ [r]}\) be the Hilbert scheme of 0-dimensional subschemes of \(S\) of length \(r\) and let \(\text{Grass}(r,\Gamma (L))\) be the Grassmannian of all \(r\)-dimensional quotients of \(\Gamma(L)\). Then the rational map \[ \varphi_k: S^{[k+1]}\to \text{Grass} \bigl(k+1, \Gamma(L)\bigr), \] sending \((Z,{\mathcal O}_Z)\in S^{[k+1]}\) into the quotient \(\Gamma(L) \to\Gamma ({\mathcal O}_Z(L))\), is indeed a morphism, and particularly it is an embedding if and only if \(L\) is \((k+1)\)-very ample.
We study \(k\)-very ample and \(k\)-spanned line bundles on complex abelian surfaces. We give the numerical criterion for an ample line bundle \(L\) to be \(k\)-very ample (resp. \(k\)-spanned) for \(k\geq 0\), which implies the results of T. Bauer and T. Szemberg [Trans. Am. Math. Soc. 349, No. 4, 1675-1683 (1997; Zbl 0952.14005)]. Our first main result is the following theorem.
Theorem 1.1. Let \(L\) be an ample line bundle on an abelian surface \(A\) and let \(k\) be a nonnegative integer. Then the following conditions are equivalent:
(1) \(L\) is \(k\)-very ample;
(2) \(L\) is \(k\)-spanned;
(3) \(L^2\geq 4k+6\) and there exists no effective divisor \(D\) satisfying the inequalities \[ 2\sqrt{(2k+3) (p_a(D)-1)} \leq L\cdot D\leq 2p_a(D) +k-1\leq 2k+1, \] where \(p_a(D)\) is the arithmetic genus of \(D\).
Note that, from this theorem, we obtain that \(L^2\geq 4k+6\) for any \(k\)-very ample line bundle \(L\) on an abelian surface. – S. Ramanan [Proc. Lond. Math. Soc., III. Ser. 51, 231-245 (1985; Zbl 0603.14013)] gave a numerical criterion for an ample line bundle on an abelian surface to be very ample under the condition that the surface does not contain elliptic curves. We deduce his result from theorem 1.1.
Our second result is a classification of \(k\)-very ample line bundles on abelian surfaces by their types.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14K10 Algebraic moduli of abelian varieties, classification
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14E25 Embeddings in algebraic geometry
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References:

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