On the projective normality of double coverings over a rational surface (Q330161)

Let \(S\) be a smooth rational surface with \(\dim|-K_S| \geq 1\) and \(\pi : X\to S\) a double covering with \(X\) minimal (possibly singular). Let \(L\) be a line bundle on \(S\) such that \(K_S+L\) is nef and \(L\cdot C\geq 3\) for every curve \(C\subset S\). The authors prove that \(K_X+\pi ^\ast L\) is spanned and the maps \(\mathrm{Sym}^r(H^0(K_X+\pi ^\ast L)) \to H^0(r(K_X+\pi ^\ast (L))\) are surjective for all \(r>0\). This theorems applies for instance to Horikawa surfaces and to certain \(K3\) surfaces. As a corollary they get that if \(X\) is smooth and \(A\) is an ample line bundle on \(S\), then \(K_X+r\pi ^\ast (A)\) is very ample and projectively normal for all \(r\geq 3\).