Zariski’s conjecture and Euler–Chow series

Abstract

We study the relations between the finite generation of Cox ring, the rationality of Euler–Chow series and Poincaré series and Zariski’s conjecture on dimensions of linear systems. We prove that if the Cox ring of a smooth projective variety is finitely generated, then all Poincaré series of the variety are rational. We also prove that the multi-variable Poincaré series associated to big divisors on a smooth projective surface are rational, assuming the rationality of multi-variable Poincaré series on curves.

Appendix A: Examples of \(E_X\)

The Euler–Chow series is in general very hard to compute. There are however a few examples where it is rational. Here we write two known examples, for details see [4]. In there, it is proved that the invariant subvarieties under the torus action are in bijection with the cones of the Fan associated to the toric varieties. Let ${\mathcal {C}}_{\lambda }$ be the Chow variety of effective cyles with homology class equal to $\lambda \in H_{2p}(X,{\mathbb {Z}})$, and ${\mathcal {C}}_{\lambda }^T$ the subset of fixed points under the torus action. It is well known that $\chi ({\mathcal {C}}_{\lambda })=\chi ({\mathcal {C}}_{\lambda }^{T}).$

1.1 Appendix A1: Projective space ${\mathbb {P}}^n$

Let $X = \mathbf{P}^{n}$ be the complex projective space of dimension n. Let $\{e_{1},\ldots ,e_{n}\}$ be the standard basis for $\mathbf{R}^{n}$. Consider $A \, = \, \{e_{1},\ldots ,e_{n+1}\}$ a set of generators of the fan $\Delta$, where $e_{n+1} = - \sum _{i=1}^{n} e_{i}$. We have the following equality

$$\begin{aligned} H^{*}(X, \, \mathbf{Z}) \, \cong \, \mathbf{Z} \, [t_{1}, \ldots , t_{n+1}] \, /I \end{aligned}$$

where I is the ideal generated by

$$\begin{aligned} (i) \qquad t_{1} \cdots t_{n+1} \end{aligned}$$

and

$$\begin{aligned} (ii) \qquad t_i \sim t_j \end{aligned}$$

Therefore

$$\begin{aligned} H^{*} \, (X, \, \mathbf{Z}) \, = \, \mathbf{Z} \, [t] \, / t^{n+1}. \end{aligned}$$

Consequently, any two cones of dimension $n-p$ represent the same element in cohomology, and

$$\begin{aligned} {\displaystyle \prod _{i=1}^{(_{n-p}^{n+1})} {\left( \frac{1}{1-t} \right) } = \left( \frac{1}{1-t} \right) ^{(_{n-p}^{n+1})} = \left( \frac{1}{1-t} \right) ^{(_{p+1}^{n+1})} = E_{p}(X)}. \end{aligned}$$

1.2 Appendix A2: Hirzebruch surfaces

A set of generators for the fan $\Delta$ that represents the Hirzebruch surface $X(\Delta )$ is given by $\{e_{1}, \ldots , e_{4} \}$ with $\{e_{1}, e_{2}\}$ the standard basis for $\mathbf{R}^{2}$, and $e_{3} \, = \, -e_{1} + ae_{2}, \; \; a 1$ and $e_{4} \, = \, -e_{2}$. With the same notation as in the last examples, we have

$$\begin{aligned} H^{*} (X(\Delta )) \, = \, \mathbf{Z}[t_{1}, \ldots , t_{4}] \, / \, I \end{aligned}$$

where I is generated by

$$\begin{aligned} (i) \qquad \{ t_{1}t_{3}, \, t_{2}t_{4} \} \end{aligned}$$

and

$$\begin{aligned} (ii) \qquad \{ t_{1} -t_{3}, \, t_{2}+ at_{3} -t_{4} \} \end{aligned}$$

from (ii) we have the following conditions for the $t_{i}$’s in $H^{*}(X)$

$$\begin{aligned} t_{1} \, \sim \, t_{3} \; \; \text{ and } \; \; t_{2} \, \sim \, (t_{4}-at_{3}). \end{aligned}$$

(A.1)

A basis for $H^{*} (X)$ is given by $\{ \{0\}, t_{3}, t_{4}, t_{4}t_{1} \}$   (see [8]). The Euler series for each dimension is:

1.:

Dimension 0: There are four orbits (four cones of dimension 2), and all of them are equivalent in homology.

$$\begin{aligned} E_{0} \, = \, {\left( \frac{1}{1-t}\right) }^{4} \end{aligned}$$

2.:

Dimension 1: Again, there are four orbits (four cones of dimension 1), and the relation among them, in homology, is given by  A.1. We obtain

$$\begin{aligned} E_{1} \, = \, {\left( \frac{1}{1-t_{3}}\right) }^{2} \, \left( \frac{1}{1-t_{4}}\right) \, \left( \frac{1}{1-t_{3}^{-a} t_{4}}\right) . \end{aligned}$$

3.:

Dimension 2: The only orbit is the torus itself so

$$\begin{aligned} E_{2} \, = \, \frac{1}{1-t}. \end{aligned}$$