A note on \(k\)-very ampleness of line bundles on general blow-ups of hyperelliptic surfaces (Q266535)

In the note the author studies \(k\)-very ampleness of line bundles on general blow-ups of hyperelliptic surfaces. We say that a line bundle \(L\) is \(k\)-very ample if for every \(0\)-dimensional subscheme \(Z \subset X\) of length \(k+1\) the restriction map \[ H^{0}(X,L) \rightarrow H^{0}(X, L \otimes \mathcal{O}_{Z}) \] is surjective. This condition means basically that the subschemes of length at most \(k+1\) impose independent conditions on global sections of \(L\). Recall that a hyperelliptic surface \(S\) is a surface with Kodaira dimension \(0\) and irregularity \(q(S) = 1\). These surfaces were classified at the beginning of 20-th century by \textit{F. Enriques} and \textit{F. Severi} [Acta Math. 32, 283--392 (1909; JFM 40.0684.01); ibid. 33, 321--403 (1910; JFM 41.0522.01)], and independently by \textit{G. Bagnera} and \textit{M. de Franchis} [Rom. Acc. L. Rend. (5) 16, No. 1, 492--498, 596--603 (1907; JFM 38.0650.01)]. They showed that there are seven non-isomorphic types of hyperelliptic surfaces. In [Math. Z. 203, No. 3, 527--533 (1990; Zbl 0722.14023)] \textit{F. Serrano} provided a characterization of the group classes of numerically equivalent divisors \(\mathrm{Num}(S)\) for each of the surface's type with the multiplicities of the singular fibers in each case. One says that a line bundle \(L\) on a hyperelliptic surface \(S\) is of type \((a,b)\) if \[ L \equiv a(A/ \mu) + b(\mu / \gamma)B, \] where \(A,B\) span \(\mathrm{Num}(S)\) and \(\mu\) is defined to by the least common multiple of multiplicities of the singular fibers and \(\gamma\) denotes order of the finite abelian group which action defines \(S\). Now we are ready to present the main result of the note. Main Result. Let \(S\) be a hyperelliptic surface. Let \(k\geq 2\) and \(d > (k+1)^{2}\). Consider a line bundle \(L_{S} \equiv (a,b)\) on \(S\) with \(a\geq d+2\) and \(b \geq d+2\). Let \(r\geq 2\) and denote by \(\pi: \tilde{S} \rightarrow S\) the blowing up of \(S\) at \(r\) very general points, where \[ r \leq 0.887 \cdot \frac{L^{2}_{S}}{(k+1)^{2}}. \] Then the line bundle \(L = \pi^{*}L_{S} - k \sum_{i=1}^{r}E_{i}\) is \(k\)-very ample on \(\tilde{S}\). It is worth pointing out that the provided bound \[ r \leq 0.887 \cdot \frac{L^{2}_{S}}{(k+1)^{2}} \] depends on the estimation of multi-point Seshadri constant of a line bundle on \(S\), which means that there is some room for further improvements (but just a little bit, see Remark 1).