On secant spaces to Enriques surfaces
type: Item
instance of: scholarly article
title: On secant spaces to Enriques surfaces
zbMATH Open document ID: 1204.14014
author: Andreas Leopold Knutsen
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2010-01-14T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1260369406
review text: 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\…
reviewed by: Hiroyuki Terakawa
Mathematics Subject Classification ID: 14H51
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 14J60
Mathematics Subject Classification ID: 14E25
zbMATH DE Number: 5658090
zbMATH Keywords: gonality
zbMATH Keywords: birationally \(k\)-very ample
zbMATH Keywords: linear systems
zbMATH Keywords: secant spaces
zbMATH Keywords: Enriques surfaces
zbMATH Keywords: higher order embeddings
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5658090