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Zariski Decomposition on Algebraic Surfaces

Zariski decompostion of pseudoeffective divisors on algebraic surface has many applications. In this post, we study Zariski decomposition on algebraic spaces of dimension 2 following an idea of Sakai.

Author

Affiliation

Fei Ye

QCC-CUNY

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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 1984), 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 (M. Artin 1970), we have to work in a larger 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 (Mazur 1975)

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 (Knutson 1971). Another good resource is the stack project.

1 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, (Knutson 1971), 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, (Knutson 1971)).

All 2-dimensional nonsingular algebraic spaces are schemes (see Remark 4.10, (Knutson 1971) or Theorem 2.7, (Michael Artin 1971).

By Lemma 71.9.3 of the Stack Project, we know that a Weil divisor on an algebraic space $X$ is a Weil divisor on a dense open scheme $U\subset X$. If we assume that $X$ is normal, then there is an open dense scheme $U$ containing all Weil divisors of $X$. Therefore, many results on divisors on schemes are still valid on algebraic spaces, for example The Nakai-Moishezon criterion of ampleness still works on algebraic spaces (see (Gieseker 1977)).

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

2 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 $\left(\begin{array}{c}{E}_i{E}_j\end{array}\right)$ is negative definite.

Therefore, for a $\mathbb{Q}$–divisor $D$ on $X$, we define the pullback $f^{\ast }D$ as $f^{\ast }D=\tilde{D}+\sum d_i{E}_i$, where $\tilde{D}$ is the birational preimage of $D$ and $d_i$ are rational numbers uniquely determined by the equations $$\tilde{D}{E}_j+\sum _i{d}_i{E}_i{E}_j=0.$$

Then for two $\mathbb{Q}$–divisors $D$ and $F$ on $X$, the intersection number $D\cdot F$ is defined to be the rational number $$f^{\ast }D\cdot f^{\ast }F.$$

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

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

3 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 (Badescu 2001)). However, in general, the contraction will only be an algebraic space of dimension 2, which is a famous result of Artin (see (M. Artin 1970) or (Michael Artin 1971)). In analytic setting this contraction criterion was first discovered by Grauert (Grauert 1962).

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. of 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. of the Stack Project.

4 Zariski Decomposition on Surfaces

A pair $(X,D)$ consisting of a normal proper surface $X$ and a $\mathbb{Q}$–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 $(Y,D^{\prime })$ such that $D^{\prime }=f_{\ast }D$ is nef, where $f:X\to Y$ is the composition of contractions.

Let $P=f^{\ast }{D}^{\prime }$. 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 $P\cdot C=f^{\ast }{D}^{\prime }\cdot C=0$.

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 (Badescu 2001).

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

References

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.