Global sections of a linear system

Question

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.

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...

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.

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.

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Question: 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...

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.

Also, I do not look for methods that apply to rational surfaces by looking at curves in the plane.

Answers which are general to higher dimensions are valuable too.

Answer 1

This question is actually a little bit vague. Anyway, I hope you can find the following answer useful.

One of the more general results about linear systems of curves on surfaces is the following theorem, proven by I. Reider in his paper Vector bundles of rank $2$ and linear systems on algebraic surfaces (Annals of Mathematics 127):

Theorem (Reider). Let $X$ be an algebraic surface, and $D$ be a nef and big divisor on $X$. Then

1. 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 $E^2=0$.

2. 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 $E^2=0$.

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 $X^{\textrm{can}}$.

Another application is the following result for abelian surfaces, see [Birkenhake-Lange, Complex Abelian Varieties, Chapter 10]:

Theorem. 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 $ED=2$.

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)]:

Adjunction Theorem. 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

$(A)$ $|K_X+D|= \emptyset$ if and only if

$(A1)$ $X \subset \mathbb{P}^n$ is a scroll over a curve of genus $g=g(D)$ or

$(A2)$ $X= \mathbb{P}^2$, $D=\mathcal{O}_{P^2}(1)$ or $D=\mathcal{O}_{P^2}(2)$.

$(B)$ 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

$(B1)$ $X$ is a Del Pezzo surface $($in particular $X$ is rational$\,)$ or

$(B2)$ $X \subset \mathbb{P}^n$ is a conic bundle.

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

$(1)$ $X=\mathbb{P}^2(p_1, \ldots, p_7), \quad D=6L-\sum_{i=1}^7 2E_i$,

$(2)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=6L-\sum_{i=1}^7 2E_i -E_8$,

$(3)$ $X=\mathbb{P}^2(p_1, \ldots, p_8), \quad D=9L-\sum_{i=1}^8 3E_i$,

$(4)$ $X= \mathbb{P}(\mathcal{E})$, 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 $B^2=1$.

Answer 2

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.

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)

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 $V_i=U_i\times \mathbb{C}$.

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).

Now we state the following result.

"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 $\leq b-ea+ei$ ."

(By the above relation, we also can write them in $V_1$)

Specially, we have $$h^0(aC+bF)=(a+1)(b+1)-\frac{a(a+1)}{2}e$$ and by Riemann-Roch Thm, $h^1(aC+bF)=0$.

The method is very usefull to construct double (triple) covers over Hirzbruch surfaces. For instance, you can refer to [4] or [5]

[1] S.-L. Tan, Effective Behavior on multiple linear systems, Asian Journal of Mathematics, Number 2 (2004), 287-304.

[2] R. Hartshorne, Algebraic geometry, Springer-Verlag,1977.

[3] W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces (2nd), Springer-Verlag, 2004.

[4] M. Mendes Lopes, R. Pardini, Triple canonical surfaces of minimal degree, Arxiv preprint math/9807006, 1998.

[5] G. Xiao: Surfaces fibrees en courbes de genre deux, Lecture Notes in Math. vol. l137.