In this post, we fix an algebraically closed field $$k$$ of arbitrary characteristic. An algebraic surface over $$k$$ is referred to an integral and reduced algebraic $$k$$-scheme of dimension 2.

In , Sakai proved the Zariski decomposition on a normal algebraic surface over $$\mathbb{C}$$ using the existence of contraction of connected negative definite curves. We will follow Sakai’s approach but work with $$k$$. Although contractions of connected negative definite curves still exists , we have to work a lager category, the category of algebraic spaces. Roughly speaking, an algebraic space is an analogue of an analytic space and may be viewed as > … local ringed spaces that are locally affine for the étale topology. > Joseph Mazur

In this post, all algebraic spaces are assumed to be Noetherian, separated, and of finite type over $$k$$.

A standard reference for algebraic spaces is . Another good resource is the stack project.

## Some Basic Facts about Algebraic Spaces

Although the category of algebraic spaces is larger than the category of schemes, it is not too much larger.

From Theorem 4.9, , we know that all 1-dimensional algebraic spaces are schemes.

In general, an algebraic space always has a dense open subset which is an affine scheme (see Corollary 6.8, ).

All 2-dimensional nonsingular algebraic spaces are schemes (see Remark 4.10, or Theorem 2.7, .

Combining Theorem 4.9, with Proposition 6.6 and 6.7, , we know that all Weil divisors on an algebraic space $$X$$ are Weil divisors on a dense open scheme $$U\subset X$$. By this result, results on divisors on schemes may still work on algebraic spaces, for example The Nakai-Moishezon criterion of ampleness still works on algebraic spaces (see ).

From now on, we will call a 2-dimensional algebraic space over $$k$$ a surface.

## Intersection Theorem on Surfaces

Let $$X$$ be a normal surface. Let $$f: Y\to X$$ be a resolution and $$E=\cup E_i$$ the exceptional locus of $$f$$, that is $$f(E)$$ is a finite collection of points. It is a consequence of the Hodge Index Theorem that the intersection matrix $$\begin{pmatrix}E_iE_j\end{pmatrix}$$ is negative definite.

Therefore, for a $$\QQ$$–divisor $$D$$ on $$X$$, we define the pullback $$f^*D$$ as $$f^*D=\widetilde{D}+\sum d_iE_i$$, where $$\widetilde{D}$$ is the birational preimage of $$D$$ and $$d_i$$ are rational numbers uniquely determined by the equations $\widetilde{D}E_j+\sum_{i}d_iE_iE_j=0.$

Then for two $$\QQ$$–divisors $$D$$ and $$F$$ on $$X$$, the intersection number $$D\cdot F$$ is defined to be the rational number $f^*D\cdot f^*F.$

A $$\QQ$$–divisor $$D$$ on a normal surface $$X$$ is said to be nef if $$DC\ge 0$$ for any irreducible (complete) curve $$C$$.

A $$\QQ$$–divisor $$D$$ on a normal surface $$X$$ is said to be pseudoeffective if $$D\cdot P\ge 0$$ for any nef divisor $$P$$.

## Contractions of Curves on Surfaces

Under some stronger conditions, the contraction of a connected negative definite curve on an normal algebraic surface is still a normal algebraic surface (a good reference is ). However, in general, the contraction will only be an algebraic space of dimension 2, which is a famous result of Artin (see or ). In analytic setting this contraction criterion was first discovered by Grauert .

We note that from Artin’s criterion a stronger result can be deduced.

Theorem 1 (Artin’s Contraction Criterion) Let $$X$$ be a proper and normal algebraic space of dimension 2 and $$C$$ be a connected algebraic curve on $$X$$. If the intersection matrix of $$C$$ is negative definite, then there is a proper morphism $$f: X\to Y$$ such that $$Y$$ is also proper and normal, $$f(C)$$ is a point and $$f: X\setminus f^{-1}(p)\to Y\setminus\{p\}$$ is an isomorphism.

Conversely, if $$C$$ on $$X$$ can be contracted by a proper morphism $$f: X\to Y$$, then $$C$$ is negative definite.

The reason that $$Y$$ is proper follows from the fact that $$X$$ is proper, $$Y$$ is separated and of finite type, and $$f$$ is surjective (see Lemma 65.40.7. in the Stack Project).

The reason that $$Y$$ is normal follows from the Zariski’s Main Theorem and the fact that $$C$$ is connected.

The reason that we can assume $$X$$ is normal is due to the existence of resolution of singularities of algebraic spaces of dimension 2.

Theorem 2 (Lipman-Artin’s Resolution of 2-dimensional Singularities) Let $$Y$$ be a two dimensional integral Noetherian algebraic space over $$k$$. Then $$Y$$ has a resolution of singularities by normalized blowups.

For a proof, see Theorem 87.8.3. in the Stack Project.

## Zariski Decomposition on Surfaces

A pair $$(X, D)$$ consisting of a normal proper surface $$X$$ and a $$\QQ$$–divisor is called a normal pair. An irreducible curve $$C$$ on $$X$$ is said to be an exceptional curve of first kind (w.r.t $$D$$) if $$D\cdot C<0$$ and $$C^2<0$$.

Theorem 3 (Zariski Decomposition) Let $$X$$ be a normal proper surface and $$D$$ a pseudoeffective divisor on $$X$$. Then there exists a decomposition $D=P+N$ such that 1. $$P$$ is nef, 2. either $$N=0$$ or $$N$$ is negative definite and $$P\cdot C=0$$ for any irreducible component $$C$$ of $$N$$.

Proof. Because a contraction of an exceptional curve of the first kind decreases the Picard number by 1, and the Picard number of $$X$$ is finite. By finitely many successive contractions of exceptional curves of the first kind, we obtain a normal pair $$(Y, D')$$ such that $$D'=f_*D$$ is nef, where $$f: X\to Y$$ is the composition of contractions.

Let $$P=f^*D'$$. Then $$P$$ is nef and $$N=D-P\ge 0$$. If $$D-P=0$$, we are done. Otherwise, for each component $$C$$ of $$D-P$$, by the projection formula, we know that $$P\cdot C=f^*D'\cdot C=0$$.

The fact that $$N$$ is negative definite follows from Artin’s contraction criterion.

Following Sakai, one can also consider minimal model theory of normal 2-dimensional proper algebraic spaces.

We remark that there are other proofs of the Zariski decomposition on algebraic surfaces over an algebraically closed field of any characteristic. A very good reference for algebraic surface over algebraically closed field is Bădescu’s book on algebraic surface .

In higher dimensions, it is known that a Zariski decomposition of pseudoeffective divisor may not exist.

Artin, M. 1970. “Algebraization of Formal Moduli: II. Existence of Modifications.” Annals of Mathematics 91 (1): 88–135. https://doi.org/dw8s45.
Artin, Michael. 1971. Algebraic Spaces. Yale University Press, New Haven, Conn.-London.
Badescu, Lucian Silvestru. 2001. Algebraic Surfaces. Universitext. New York: Springer-Verlag. https://doi.org/10.1007/978-1-4757-3512-3.
Gieseker, David. 1977. “On Two Theorems of Griffiths About Embeddings with Ample Normal Bundle.” American Journal of Mathematics 99 (6): 1137–50. https://doi.org/dg72d8.
Grauert, Hans. 1962. “Über Modifikationen Und Exzeptionelle Analytische Mengen.” Mathematische Annalen 146: 331–68. https://doi.org/bm8kgb.
Knutson, Donald. 1971. Algebraic Spaces. Lecture Notes in Mathematics, Vol. 203. Springer-Verlag, Berlin-New York.
Mazur, Joseph. 1975. “Conditions for the Existence of Contractions in the Category of Algebraic Spaces.” Transactions of the American Mathematical Society 209: 259–65. https://doi.org/d6xgtn.
Sakai, Fumio. 1984. “Weil Divisors on Normal Surfaces.” Duke Math. J. 51 (4): 877–87. https://doi.org/10.1215/S0012-7094-84-05138-X.

### Author's bio

Fei Ye (https://yfei.page) is an assistant professor at QCC-CUNY.

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Text and figures are licensed under Creative Commons Attribution CC BY-NC-SA 4.0.

### Citation

Fei Ye (2020). Zariski Decomposition on Algebraic Surfaces. Fei Ye's Math Blogs. /post/2020/12/12/zariski_decomposition/
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