Integral Zariski Decomposition

Algebraic Geometry

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

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September 27, 2024

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

Definition 1 Let D be a pseudo-effective divisor on a normal complete surface X. A decomposition D=PZ+NZ is said to be an integral Zariski decomposition if the following conditions hold:

  1. PZ is a Z-positive divisor on X.
  2. NZ=0 or NZ>0 is a negative definite divisor on X.
  3. PZC0 for any irreducible component of NZ.

Definition 2 Let D be a R–divisor on a normal complete surface X. We say that D is Z–positive if BD 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 (BD)C<0.

The following is a useful characterization of Z–positive divisors.

Proposition 1 (Enokizono (), Proposition 3.16) Let D be a pseudo-effective R–divisor on a normal complete surface X with the Zariski decomposition D=P+N. Then D is Z–positive if there is a chain D0=DN<D1<<Dk=D such that Ci:=DiDi1 is an irreducible reduced curve and Di1Ci>0. In particular, P is Z–positive divisor.

The reason that P is 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 (), Theorem 1.2, which should be attributed to Sakai (), we can construct an integral Zariski decomposition for any pseudo-effective R-divisor.

Theorem 1 Let D be a pseudo-effective 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=PZ+NZ with 0NZN, where PZ is Z–positive.

Proof. Write E0=N and D0=DN.

If E0=0, then D=D0 is an integral Zariski decomposition.

Suppose that E00.

If D0C0 for any irreducible component C of E0, then (CD0)C=C2D0C<0 for any irreducible curve C with C2<0. Therefore, D=D0+E0 is an integral Zaiski decomposition.

We now construct a chain of divisors inductively as follows.

If C0 be an irreducible component E0 such that D0C0>0, then we let D1=D0+C0 and E1=E0C0. If E1=0 or E1 has no irreducible component C such that D1C>0, then we let PZ=D1 and NZ=E1. Otherwise, repeating this procedure until we get a chain of R–divisors D0<D1<<Dk such that Ci=Di+1Di is an irreducible curve, Di+1Ci>0, and Ek=0 or DkC0 for any irreducible component C of Ek. Let PZ=Dk and NZ=Ek.

We claim that PZ is Z–positive.

By the construction of Dk, we know that Dk=DEk=P+NEk. Since 0EkN, Dk=P+Nk is a Zariski decomposition of Dk, where Nk=NEk0. Therefore, by , PZ=Dk is Z–positive.

We can then conclude that D=Dk+Ek 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 (LAquila, 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 Q-Divisors on Surfaces and Applications.” Asian J. Math. 22 (4): 761–85. https://doi.org/10.4310/AJM.2018.v22.n4.a8.