On secant spaces to Enriques surfaces (Q1049905)

A line bundle \(L\) on a smooth projective variety \(X\) is said to be birationally \(k\)-very ample if there exists a Zariski-open, dense subset \(U\) of \(X\) such that the restriction map \(H^0(L) \rightarrow H^0(L \otimes \mathcal{O}_Z)\) is surjective for any \(0\)-dimensional subscheme \(Z\) of \(X\) of length \(h^0(\mathcal{O}_Z)=k+1\) with \(\mathrm{Supp}(Z)\subset U\). Birationally \(k\)-very ampleness is a sort of higher order embedding, introduced in the author's paper [Manuscr. Math. 104, No. 2, 211--237 (2001; Zbl 1017.14015)]. If \(L\) is base point free, then \(L\) is birationally \(1\)-very ample if and only if the morphism \(\varphi_L\) associated to \(L\) is birational. In the paper under review, the author studies the relation between the minimal gonality of smooth curves in a complete linear system on an Enriques surface and the embedding properties of the adjoint linear system. The main result in the paper is the following. Theorem. Let \(L\) be an ample, globally generated line bundle on an Enriques surface \(S\) such that \(L^2 \geq 10\) and \(k \geq 1\) an integer. Then the following conditions are equivalent: (i) \(L+K_S\) is birationally \(k\)-very ample. (ii) The natural restriction map \(H^0(L+K_S)\rightarrow H^0((L+K_S)\otimes \mathcal{O}_Z)\) is surjective for all \(0\)-dimensional subschemes \(Z\subset S\) of length \(\leq k+1\) satisfying \(Z\cap \Theta_{k+1}(L)=\emptyset\). (iii) \(\varphi_{L+K_S}\) is birational, and if \(k \geq 2\) then \(\varphi_{L+K_S}(S)\) has no \((k+1)\)-secant \((k-1)\)-plane \(\Pi\) such that \(\Pi \cap \varphi_{L+K_S}(\Theta_{k+1}(L))=\emptyset\). (iv) All the smooth curves in \(|L|\) have gonalities \(\geq k+2\). Here for any integer \(s>0\) and any big and nef line bundle \(L\) on \(S\), \(\Theta_s(L)\) is defined to be the set \(\{ x\in S | x\) is a point of a smooth irreducible rational curve such that \(R\cdot L \leq s-2\), or a point of a reduced curve such that \(|2E|\) is an elliptic pencil and \(E\cdot L \leq s \}\). Note that \(\Theta_s(L)\) is a proper, closed subset of \(S\).