Positive curves in polarized manifolds
Curves \(Y\) on a smooth complex polarized variety \((X,A)\) of dimension \(k \geq 3\) are studied in order to generalize a paper by \textit{R. Paoletti} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 6, No. 4, 259-274 (1995; Zbl 0874.14018)] in case \(k=3\) about the Seshadri constant \(\varepsilon (Y,A)\) and positivity conditions on \(Y\). Let \(\pi : X_Y \rightarrow X\) be the blow up of \(X\) at the smooth curve \(Y\), \(E\) the exceptional divisor of the blow up, then the Seshadri constant is defined by: \(\varepsilon := \varepsilon (Y,A) := \sup\{\eta \in {\mathbb{Q}}\mid \pi ^* A-\eta E\) is ample\}. Then for all \(\eta \in (0,\varepsilon)\cap {\mathbb{Q}}\), \(\eta = {n\over m}\), we can assume \(m(\pi^* A)- nE\) to be very ample, so, if \(H_1,\dots,H_{k-2}\) are generic elements in \(m(\pi^* A)- nE\), \({\mathcal H} = H_1\cap \dots \cap H_{k-2}\) and \(Y' = {\mathcal H}\cap Y\) will be smooth and connected (of dimension 2 and 1 respectively). The curve \(Y\) is said to be \(A\)-ample if it exists \(\eta \in (0,\varepsilon)\) such that \(Y'\) is ample in \({\mathcal H}\), and is said to be A-big if \({Y'}^2 > 0\). The main results in the paper are: -- any \(A\)-ample curve \(Y\) meets every hypersurface of \(X\), and any \(A\)-big curve meets every hypersurface on \(X\) which is smooth and with an ample normal bundle; -- an \(A\)-big curve \(Y\) is not \(A\)-ample if and only if there is an irreducible hypersurface in \(X\) which does not meet \(Y\); -- when \(Y\) is \(A\)-ample or \(A\)-big, criteria to compute the cohomological dimension of \(X-Y\) are given; -- \(A\)-ampleness and \(A\)-bigness are open properties in flat families.
scientific article; zbMATH DE number 1002960
scientific article