On the canonical ring of a surface
Kazuhiro Konno (Osaka University)”Singularity Theory and Commutative Algebra”, September 12th–16th, 2005

Let S be a non-singular, projective, minimal surface of general type defined over the complex
number field C. The canonical ring of S is of course the graded C-algebra
R(S , KS ) =+∞M

H 0 (S , mKS ).

m=0

This naive object has been studied by several authors, after the success of Kodaira and Bombierion the pluri-canonical systems. Let me recall some important results obtained so far: R(S , KS ) is
• generated in degrees ≤ 5, when pg (S ) > 0 (Ciliberto [1], ’83),• generated in degrees ≤ 3 and related in degrees ≤ 6, when pg (S ) ≥ 2, q(S ) = 0, KS2 ≥ 3 andthe canonical model has an irreducible canonical curve (Reid [7], ’88),• generated in degrees ≤ 4, when pg (S ) = 0 and 2KS is free (Mendes Lopes [6], ’97).
The difficulty arises from the simple fact that |KS | may well have base points or fixed components,unlike curves. Hence we must accept it as a common rule that R(S , KS ) needs a generator in degree
3. Then it is too much to hope that the presence of a primitive generator in “high” degree forces
S to have a “special” linear system. This may be enough to disappoint everyone. However, it is
not a tiny thing to know how the algebraic structure of R(S , KS ) reflects on the geometry of S , andvice versa.
I talk about my recent results on the degree bounds for generators and relations. A remarkable
theorem essentially due to Francia [3] and Reider [8] states that 2KS is free except possibly whenpg (S ) = 0 and KS2 ≤ 4. My first result is a simple application of the theorem via the Koszul
cohomology [2], and is nothing more than what one naturally expects after Ciliberto’s bound forgenerators.
Theorem 1 Let S be a minimal algebraic surface of general type such that 2KS is free. Then
R(S , KS ) is generated in degrees ≤ 5 and related in degrees ≤ 10. If furthermore q(S ) = 0, thenR(S , KS ) is generated in degrees ≤ 4 and related in degrees ≤ 8 except when (pg (S ), KS2 ) = (2, 1).
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A numerically 2-connected curve E is called a Francia cycle if either (i) pa (E) = 1, E 2 = −1,or (ii) pa (E) = 2, E 2 = 0. It is known that R(S , KS ) needs a generator in degree 4 (and a relationin degree 8) if the fixed part of |KS | contains a Francia cycle. Many such surfaces are known toexist. Hence Theorem 1 cannot be improved further (at least for regular surfaces) in this sense.
Nevertheless, it is an interesting problem to detect whether any criminal is a Francia cycle in thefixed part, as in the case of fibrations [4] (see also Theorem 5 below). Here is my partial answer:
Theorem 2 Let S be a minimal algebraic surface of general type with pg (S ) ≥ 2, q(S ) = 0 andKS2 ≥ 3. Let |KS | = |M| + Z be the decomposition into its variable and fixed parts, and supposethat |M| has an irreducible member. If one of the following conditions (1), (2) is satisfied, thenR(S , KS ) is generated in degrees ≤ 3 and related in ≤ 6.
(1) H 0 (Z, KZ ) = 0 (possibly Z = 0).
(2) Z does not contain any Francia cycles and decomposes as Z = ∆ + Γ1 + · · · + Γn , where(a) ∆ is an effective divisor with KS ∆ = 0 (possibly ∆ = 0), Supp(∆) ∩ Supp(Z − ∆) = ∅, and(b) for each i ∈ {1, . . . , n}, Γi is a chain connected curve such that KS Γi > 0, OΓi (−Γi ) is nef,
Γ2 ≤ 0 holds for any subcurve Γ  Γi and, when j , i, OΓi (Γ j ) is numerically trivial.
My proof depends on the hyperplane-section principle which does not work well without assumingq(S ) = 0. |M| contains an irreducible member unless the canonical map is composed of a pencil;
so it may be harmless to assume it. If Z supports exceptional sets of rational singular points at
most, then h0 (Z, KZ ) = 0; so (1) is an extension of Reid’s theorem. An effective divisor D ischain connected if either it is irreducible, or for any proper subcurve Γ there exists an irreducible
component C of Γ such that (D − Γ)C > 0; hence the fundamental cycle of a normal surface
singularity is automatically chain connected and serves an example of Γi in (b). Though I do hopethat the technical assumptions in (2) will be removed eventually, it would be worth stating here
the following theorem which is crucial in the proof of Theorem 2, and which is a generalizationof a theorem in [5] to the negative semi-definite case.
Theorem 3 Let D be a chain connected curve on a non-singular surface S with KS2 > 0 such that
KS and −D are both nef on D. Assume furthermore that ∆2 ≤ 0 holds for any subcurve ∆ of D.
Let Lα (α = 1, 2) be a line bundle on D such that Lα − 2KS is nef. Then the multiplication map
H 0 (D, L1 ) ⊗ H 0 (D, L2 ) → H 0 (D, L1 + L2 )is surjective, unless there exists a subcurve E  D such that L1 ≡ L2 ≡ 2KS on E and either
pa (E) = 1, E 2 = −1, or pa (E) = 2, E 2 = 0. Here, the symbol ≡ means the numerical equivalence.
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The method also applies to the local and the semi-global cases for bounding the degrees of relations. The following two results answer Reid’s 1–2–3 conjecture for the corresponding algebras.
As is well-known, the bound for generators is due to Laufer [5] in the case of singularities.
Theorem 4 The relative canonical algebra for a normal surface singularity is generated in degrees ≤ 3 and related in degrees ≤ 6.
Theorem 5 The relative canonical algebra for a relatively minimal fibration of curves of genus≥ 2 is generated in degrees ≤ 4 and related in degrees ≤ 8. If furthermore there are no multiple
fibres whose canonical system has a (−1) elliptic cycle as a fixed component, then the algebra isgenerated in degrees ≤ 3 and related in degrees ≤ 6.

References[1] C. Ciliberto, Sul grado dei generatori dell’anello canonico di una superficie di tipo generale, Rend.
Sem. Mat. Univ. Politec. Torino 41:3 (1983/84), 83–111[2] M. Green, Koszul cohomology and geometry of projective varieties, J. Diff. Geom. 19 (1984), 125–171[3] P. Francia, On the base points of the bicanonical system, Problems in the theory of surfaces and theirclassification (Cortona, 1988), pp. 141–150, Sympos. Math., XXXII, Academic Press, London, 1991
[4] K. Konno, 1-2-3 theorem for curves on algebraic surface, J. reine. angew. Math. 533 (2001), 171–205[5] H.B. Laufer, Generation of 4-pluricanonical forms for surface singularities, Amer. J. Math. 109(1987), 571–589[6] M. Mendes Lopes, The degree of the generators of the canonical ring of surfaces of general type withpg = 0. Arch. Math. 69, no. 5 (1997), 435–440[7] M. Reid, Infinitesimal view of extending a hyperplane section – deformation theory and computer
algebra, Lect. Notes in Math. 1417, Algebraic Geometry Proceedings, L’Aquila 1988, A.J.Sommeseet. al. eds. 1990, Springer-Verlag, pp. 214–286[8] I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces, Ann. of Math. 127 (1988),
309–316

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