Local Positivity of Ample Line Bundles
| dc.creator | Ein, Lawrence | |
| dc.creator | Küchle, Oliver | |
| dc.creator | Lazarsfeld, Robert | |
| dc.date | 1994-08-09 | |
| dc.date.accessioned | 2026-07-07T09:06:08Z | |
| dc.date.available | 2026-07-07T09:06:08Z | |
| dc.description | Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of dimension $n$. Demailly has introduced a very interesting invariant --- the Seshadri constant $ε(L,x)$ --- which in effect measures how positive $L$ is locally near a given point $x \in X$. For instance, Seshadri's criterion for ampleness may be phrased as stating that $L$ is ample if and only if there exists a positive number $e > 0$ such that $ε(L,x) > e$ for all $x \in X$, and if $L$ is VERY ample, then $ε(L,x) \ge 1$ for every $x$. We prove the somewhat surprising result that in each dimension $n$ there is a uniform lower bound on the Seshadri constant of an ample line bundle $L$ at a very general point of $X$. Specifically, $ε(L,x) \ge (1/n) $ for all $x \in X$ outside the union of countably many proper subvarieties of $X$. Examples of Miranda show that there cannot exist a bound (independent of $X$ and $L$) that holds at every point. The proof draws inspiration from two sources: first, the arguments used to prove boundedness of Fano manifolds of Picard number one; and secondly some of the geometric ideas involving zero-estimates appearing in the work of Faltings and others on Diophantine approximation and transcendence theory. We give some elementary applications of the main theorem to adjoint and pluricanonical linear series. | |
| dc.description | 23 pages, AMS-TeX 2.1 | |
| dc.identifier | https://arxiv.org/abs/alg-geom/9408003 | |
| dc.identifier | http://arxiv.org/abs/alg-geom/9408003 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149912 | |
| dc.subject | Algebraic Geometry | |
| dc.title | Local Positivity of Ample Line Bundles | |
| dc.type | text |