On the irreducibility of Severi varieties on \(K3\) surfaces. (English) Zbl 1423.14192
Let \((X,L)\) be a polarized \(K3\) surface of genus \(p\ge 11\) with \(L\) the generator of \(\mathrm{Pic}(S)\). The Severi variety \(V^{L,\delta}\) is the parameters spaces for the set of all integral nodal curves \(D\in |L|\) with exactly \(\delta\) nodes. Here the authors prove that \(V^{L,\delta}\) is irreducible if \(p\ge 4\delta -3\) improving in the case of the principal polarization a theorem by M. Kemeny [Bull. London Math. Soc. 43, No. 1, 159–174 (2013; Zbl 1032.14005)], which say that for any \(k\ge 1\) \(V^{kL,\delta}\) is irreducible if \(\delta < \frac{6(2p-2)+8}{(11(2p-2)+12)^2}k^2(2p-2)^2\).
Reviewer: Edoardo Ballico (Povo)
Citations:
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