Global sections of a linear system - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:00:27Z http://mathoverflow.net/feeds/question/70106 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/70106/global-sections-of-a-linear-system Global sections of a linear system Jesus Martinez Garcia 2011-07-12T11:31:24Z 2011-07-12T13:22:14Z <p>Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of several linear systems. For instance, if $h$ is a section of $\mathbb{P}^1$ and $f$ is a fibre, then I would be looking for $h^0(\mathbb{F}_n,\vert h+kf\vert),\ k\geq 0$. The reason to do this was to find morphisms into projective space.</p> <p>Also I was interested in knowing the usual stuff: whether the linear system had fixed points, whether it separated points and tangents, in which points it didn't... </p> <p>My approach was to do it for $\mathbb{F}_0\cong \mathbb{P}^1\times \mathbb{P}^1$ and then by induction find it for the rest of $\mathbb{F}_n$ using elementary transformations.</p> <p>It seemed to me a bit 'ad hoc' and I feel the only reason I could do this is because I had a very explicit knowledge of $P^1\times P^1$ and how to find the other surfaces from this one. If I had started with the image via that linear system into projective space, even with lots of information about it, I doubt I had been able to find so much information or even understand which curves were linearly equivalent.</p> <hr> <p><strong>Question</strong>: Are there methods to find information (dimension, base points, incidence) about linear systems of divisors in a surface given (some) explicit information about the geometry of that surface? By information I mean configuration of lines, degree, whether it has curves embedded inside, intersection of particular curves...</p> <p>I am aware this is a bit of a vague question, but that is precisely the point, I do not seek solutions to particular examples but tools that work for as many surfaces as possible.</p> <p>Also, I do not look for methods that apply to rational surfaces by looking at curves in the plane.</p> <p>Answers which are general to higher dimensions are valuable too.</p> http://mathoverflow.net/questions/70106/global-sections-of-a-linear-system/70111#70111 Answer by Francesco Polizzi for Global sections of a linear system Francesco Polizzi 2011-07-12T12:18:49Z 2011-07-12T12:35:51Z <p>This question is actually a little bit vague. Anyway, I hope you can find the following answer useful.</p> <p>One of the more general results about linear systems of curves on surfaces is the following theorem, proven by I. Reider in his paper <a href="http://www.jstor.org/stable/2007055" rel="nofollow">Vector bundles of rank $2$ and linear systems on algebraic surfaces (Annals of Mathematics 127)</a>:</p> <blockquote> <p><strong>Theorem (Reider).</strong> Let $X$ be an algebraic surface, and $D$ be a nef and big divisor on $X$. Then </p> <ol> <li><p>If $D^2 \geq 5$ and $x$ is a base point of $|K_X+D|$, then there exists a curve $E$ on $X$ with $x \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $DE=1$ and $E^2=0$.</p></li> <li><p>If $D^2 \geq 10$ and $x,y$ are two points, possibly infinitely near, such that $|K_X + D|$ does not separate $x$ and $y$, then there exists a curve $E$ on $X$ with $x,y \in \operatorname{Supp} E$ such that either $DE=0$ and $E^2=-1$ or $-2$ or $DE=1$ and $E^2=0$ or $1$ or $DE=2$ and $E^2=0$.</p></li> </ol> </blockquote> <p>This result has many important consequences. For instance, it can be used to deduce Bombieri's theorem for pluricanonical systems (if $X$ is a surface of general type, then the $5$-canonical map is a birational morphism of $X$ onto its canonical model $X^{\textrm{can}}$.</p> <p>Another application is the following result for abelian surfaces, see [Birkenhake-Lange, Complex Abelian Varieties, Chapter 10]:</p> <blockquote> <p><strong>Theorem.</strong> Suppose $D$ is an ample line bundle of type $(1,d)$, with $d \geq 5$, on an abelian surface $X$. Then the morphism $$\varphi_D \colon X \longrightarrow \mathbb{P}^{d-1}$$ is an embedding if and only if there is no elliptic curve $E$ on $X$ with $ED=2$. </p> </blockquote> <p>When $D$ is very ample, one can also use the following classical result, known as "adjunction theorem" and whose modern form is due to Sommese [Hyperplane sections of projective surfaces I - The adjunction mapping. Duke Math. J. 46 (1979)]:</p> <blockquote> <p><strong>Adjunction Theorem.</strong> Let $X \subset \mathbb{P}^n$ be a smooth surface and $D$ the hyperplane class. Then $|K_X+D|$ is not special and has dimension $N=g(D)+p_g(X)-q(X)-1$. Moreover</p> <p>$(A)$ $|K_X+D|= \emptyset$ if and only if</p> <p>$(A1)$ $X \subset \mathbb{P}^n$ is a scroll over a curve of genus $g=g(D)$ or </p> <p>$(A2)$ $X= \mathbb{P}^2$, <code>$D=\mathcal{O}_{P^2}(1)$</code> or <code>$D=\mathcal{O}_{P^2}(2)$</code>.</p> <p>$(B)$ If $|K_X+D| \neq \emptyset$ then $|K_X+D|$ is basepoint free. In this case $(K_X+D)^2=0$ if and only if</p> <p>$(B1)$ $X$ is a Del Pezzo surface $($in particular $X$ is rational$\,)$ or </p> <p>$(B2)$ $X \subset \mathbb{P}^n$ is a conic bundle. </p> <p>If $(K_X+D)^2>0$ then the map $$\varphi_{|K_X+D|} \colon X \longrightarrow X' \subset \mathbb{P}^N$$ defined by $|K_X+D|$ is birational onto a smooth surface $X'$ of degree $(K_X+D)^2$ and blows down all the lines $E$ on $X$ such that $K_XE=-1$, unless</p> <p>$(1)$ $X=\mathbb{P}^2(p_1, \ldots, p_7), \quad D=6L-\sum_{i=1}^7 2E_i$, </p> <p>$(2)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=6L-\sum_{i=1}^7 2E_i -E_8$, </p> <p>$(3)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=9L-\sum_{i=1}^8 3E_i$,</p> <p>$(4)$ $X= \mathbb{P}(\mathcal{E})$, where $\mathcal{E}$ is an indecomposable rank $2$ vector bundle over an elliptic curve and $D=3B$, where $B$ is an effective divisor on $X$ with $B^2=1$.</p> </blockquote> http://mathoverflow.net/questions/70106/global-sections-of-a-linear-system/70115#70115 Answer by Jun Lu for Global sections of a linear system Jun Lu 2011-07-12T13:22:14Z 2011-07-12T13:22:14Z <p>As I know, Sheng-li Tan is an expert on the linear systems on algebraic surfaces. You can refer to his papers such as [1]. I think you will find what you desire.</p> <p>In the case for ruled surface, you can also refer to [2] (Ch. V, Sec.2, page 369) or [3](Ch.V, Sec. 4,page,189)</p> <p>Now I will show you how to find all sections of $|aC+bF|$ on $\mathbb{F}_e$ where $C$ is a section with $C^2=-e$ and $F$ is a fiber. For convinience, we can assume $b\geq ea> 0$. Let $[t_0:t_1]$ be the coordinates of base curve $\mathbb{P}^1$, $U_i={[t_0:t_1]\mid t_i\ne 0}$ be the affine covers ($i=0,1$). Thus we have an affine cover $\mathbb{F}_e-C=V_1\cup V_2$ where $V_i=U_i\times \mathbb{C}$. </p> <p>One can consider the local coordinaters $(t,u)$ on V_1 and $(s,v)$ on $V_2$. The relation is $$t=\frac{1}{s}, u=\frac{v}{s^e}.$$ (You can get it by considering $\mathbb{F}_e$ as a projective bundle).</p> <p>Now we state the following result.</p> <p>"Each section of $|aC+bF|$ can be represented as the following local equation in $V_0$ $$\sum\limits_{i=0}^a d_i(t)u^{a-i},$$ where $d_i(t)$'s are the polynomials of degree $\leq b-ea+ei$ ."</p> <p>(By the above relation, we also can write them in $V_1$)</p> <p>Specially, we have $$h^0(aC+bF)=(a+1)(b+1)-\frac{a(a+1)}{2}e$$ and by Riemann-Roch Thm, $h^1(aC+bF)=0$.</p> <p>The method is very usefull to construct double (triple) covers over Hirzbruch surfaces. For instance, you can refer to [4] or [5]</p> <p>[1] S.-L. Tan, Effective Behavior on multiple linear systems, Asian Journal of Mathematics, Number 2 (2004), 287-304. </p> <p>[2] R. Hartshorne, Algebraic geometry, Springer-Verlag,1977.</p> <p>[3] W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces (2nd), Springer-Verlag, 2004.</p> <p>[4] M. Mendes Lopes, R. Pardini, Triple canonical surfaces of minimal degree, Arxiv preprint math/9807006, 1998.</p> <p>[5] G. Xiao: Surfaces fibrees en courbes de genre deux, Lecture Notes in Math. vol. l137. </p>