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Let X be an smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete linear systems associated to powers of the canonical class $nK_{X}$. I would like to understand some examples of how intricate these maps can be for small $n$.

For simplicity let's restrict to the case of algebraic surfaces. We assume that $X$ is of general type and minimal. A concrete question that I have in mind is the following. One expects that for large $n$ the complete linear systems become basepoint free, but how large is large? What explicit examples are known where the systems $nK_{X}$ all have a basepoint for $n \leq N$ but become free when $n>N$ ? Is there some absolute bound on $N$, and if so for what surface is the bound saturated?

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In the general case when $n=\dim X$ is arbitrary Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case. In the situation at hand it implies that $\vert ((n+3)!)K_X\vert$ is basepoint-free (actually one can write $(n+2)(n+2)!$ instead of $(n+3)!$, but this is shorter).

See: Kollár, János Effective base point freeness. Math. Ann. 296 (1993), no. 4, 595–605.

A closely related problem is Fujita's conjecture, which states that if $L$ is an ample divisor on $X$, then $|K_X+mL|$ is basepoint-free if $m\geq \dim X+1$ and very ample if $m\geq \dim X +2$.

There are many related results, but the conjecture is still open.

Angehrn and Siu prove that if $L$ is ample, then $|mL+K_X|$ is base point free if $m\geq \frac 12(n^2+n+2)$.

Angehrn, Urban; Siu, Yum Tong Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, 291–308. 32J25 (14C20 32L10)

Helmke also has some related results:

Helmke, Stefan . The base point free theorem and the Fujita conjecture. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 215--248, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.

Helmke, Stefan . On global generation of adjoint linear systems. Math. Ann. 313 (1999), no. 4, 635--652.

Helmke, Stefan . On Fujita's conjecture. Duke Math. J. 88 (1997), no. 2, 201--216.

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Bombieri's theorem states that, for $n \geq 5$, the $n$-th pluricanonical map $S \to \mathbb{P}^N$ is a birational morphism onto its image, which contracts only the $(-2)$-curves of $S$. In particular it is everywhere defined and $|nK_X|$ is base-point free (moreover the image is isomorphic to the canonical model of $S$). Furthermore, $nK_S$ is spanned by global sections for $n \geq 4$. The main technical tool required is Ramanujam's vanishing theorem, which is not available in higher dimensions, so a straightforward generalization is not possible. –  Francesco Polizzi Nov 16 '10 at 10:09
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Francesco, you are absolutely right. I don't know what I was thinking... –  Sándor Kovács Nov 16 '10 at 15:35
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For a minimal complex surface of general type, to the best of my knowledge, the situation is as follows:

  • $|nK_X|$ is free for $n\ge 4$;

  • $|3K_X|$ is free, except possibly for $K_X^2=1$;

  • If $K^2_X=1$, then $|2K_X|$ has base points. In the remaining cases $|2K_X|$ is free, except possibly for $K^2_X=2,3,4$ and $p_g(X)=0$. (Examples with $K^2=2$, $p_g=0$ and $|2K_X|$ not free are known).

Statements 1),2) and statement 3) for $K^2_X>4$ are essentially due to Bombieri (MR0318163) and can be easily reobtained using Reider's Theorem. The proof that $|2K_X|$ is free if $p_g(X)>0$ is due to P. Francia (MR1273376) except for the case $p_g=q=1$ which was settled by Catanese-Ciliberto (MR1273372).

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What I know is that Bombieri proved that for algebraic surfaces of general type $h^0(2K_X)\not=0$ and the map defined by $|nK_X|$ is always birational for every $n\geq 5$. Moreover if you take a minimal surface $X$ with geometric genus 2 and $(K_X)^2=1$, then $|4K_X|$ is not birational.

The same questions are very interesting and widely open in higher dimension.

We know only partial results for 3-folds and 4-folds, you can see for example the introduction of http://uk.arxiv.org/PS_cache/arxiv/pdf/1001/1001.3340v1.pdf if you are interested.

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Thanks for the references! –  Clay Cordova Nov 16 '10 at 7:44
    
I think you meant that the map $\varphi_n$ defined by $|nK_S|$ is a birational morphism onto its image for $n \geq 5$. The statement is even stronger, since it says that the image of $\varphi_n$ is isomorphic to the canonical model of $S$. –  Francesco Polizzi Nov 16 '10 at 10:14
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