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Integral Zariski Decomposition

We present a construction of integral Zariski decomposition for a pseudo-effective divisor on a normal complete algebraic surface.

Author

Affiliation

Fei Ye

QCC-CUNY

In this short note, we present a constructive proof for the existence of the integral Zariski decomposition as defined in Enokizono (2024).

Definition 1 Let $D$ be a pseudo-effective divisor on a normal complete surface $X$. A decomposition $D=P_{\mathbb{Z}}+N_{\mathbb{Z}}$ is said to be an integral Zariski decomposition if the following conditions hold:

1. $P_{\mathbb{Z}}$ is a $\mathbb{Z}$-positive divisor on $X$.
2. $N_{\mathbb{Z}}=0$ or $N_{\mathbb{Z}}>0$ is a negative definite divisor on $X$.
3. $-P_{\mathbb{Z}}\cdot C\ge 0$ for any irreducible component of $N_{\mathbb{Z}}$.

Definition 2 Let $D$ be a $\mathbb{R}$–divisor on a normal complete surface $X$. We say that $D$ is $\mathbb{Z}$–positive if $B-D$ is not nef over $B$ for any negative definite integral divisor $B>0$, i.e. there exists an irreducible component $C$ of $B$ such that $(B-D)C<0$.

The following is a useful characterization of $\mathbb{Z}$–positive divisors.

Proposition 1 (Enokizono (2024), Proposition 3.16) Let $D$ be a pseudo-effective $\mathbb{R}$–divisor on a normal complete surface $X$ with the Zariski decomposition $D=P+N$. Then $D$ is $\mathbb{Z}$–positive if there is a chain $$D_0=D-\lfloor N\rfloor <D_1<\cdots <D_k=D$$ such that $C_i:=D_i-D_{i-1}$ is an irreducible reduced curve and $D_{i-1}{C}_i>0$. In particular, $\lceil P\rceil$ is $\mathbb{Z}$–positive divisor.

The reason that $\lceil P\rceil$ is $\mathbb{Z}$–positive is that the nef part of the Zariski decomposition must be at least $P$ and the negative definite part is then fractional, i.e. less than $.

Applying the method in the proof of Ye, Zhang, and Zhu (2018), Theorem 1.2, which should be attributed to Sakai (1990), we can construct an integral Zariski decomposition for any pseudo-effective $\mathbb{R}$-divisor.

Theorem 1 Let $D$ be a pseudo-effective $\mathbb{R}$–divisor on a normal complete surface $X$ and $D=P+N$ the Zariski decomposition of $D$. Then it has an integral Zariski decomposition $D=P_{\mathbb{Z}}+N_{\mathbb{Z}}$ with $0\le N_{\mathbb{Z}}\le \lfloor N\rfloor$, where $P_{\mathbb{Z}}$ is $\mathbb{Z}$–positive.

Proof. Write $E_0=\lfloor N\rfloor$ and $D_0=D-\lfloor N\rfloor$.

If $E_0=0$, then $D=D_0$ is an integral Zariski decomposition.

Suppose that $E_0\ne 0$.

If $D_{0}C\le 0$ for any irreducible component $C$ of $E_0$, then $(C-D_0)C=C^2-D_{0}C<0$ for any irreducible curve $C$ with $C^2<0$. Therefore, $D=D_0+E_0$ is an integral Zaiski decomposition.

We now construct a chain of divisors inductively as follows.

If $C_0$ be an irreducible component $E_0$ such that $D_0{C}_0>0$, then we let $D_1=D_0+C_0$ and $E_1=E_0-C_0$. If $E_1=0$ or $E_1$ has no irreducible component $C$ such that $D_{1}C>0$, then we let $P_{\mathbb{Z}}=D_1$ and $N_{\mathbb{Z}}=E_1$. Otherwise, repeating this procedure until we get a chain of $\mathbb{R}$–divisors $D_0<D_1<\cdots <D_k$ such that $C_i=D_{i+1}-D_i$ is an irreducible curve, $D_{i+1}{C}_i>0$, and $E_k=0$ or $D_{k}C\le 0$ for any irreducible component $C$ of $E_k$. Let $P_{\mathbb{Z}}=D_k$ and $N_{\mathbb{Z}}=E_k$.

We claim that $P_{\mathbb{Z}}$ is $\mathbb{Z}$–positive.

By the construction of $D_k$, we know that $D_k=D-E_k=P+N-E_k$. Since $0\le E_k\le \lfloor N\rfloor$, $D_k=P+N_k$ is a Zariski decomposition of $D_k$, where $N_k=N-E_k\ge 0$. Therefore, by Proposition 1, $P_{\mathbb{Z}}=D_k$ is $\mathbb{Z}$–positive.

We can then conclude that $D=D_k+E_k$ is an integral Zariski decomposition.

References

Enokizono, Makoto. 2024. “An Integral Version of Zariski Decompositions on Normal Surfaces.” European Journal of Mathematics 10 (2): 38. https://doi.org/10.1007/s40879-024-00750-4.

Sakai, Fumio. 1990. “Reider-Serrano’s Method on Normal Surfaces.” In Algebraic Geometry (L’Aquila, 1988), 1417:301–19. Lecture Notes in Math. Springer, Berlin. https://doi.org/10.1007/BFb0083346.

Ye, Fei, Tong Zhang, and Zhixian Zhu. 2018. “Sakai’s Theorem for $\mathbb{Q}$-Divisors on Surfaces and Applications.” Asian J. Math. 22 (4): 761–85. https://doi.org/10.4310/AJM.2018.v22.n4.a8.