[Paper Review] On pluricanonical systems of algebraic varieties of general type

Meng Chen|ArXiv.org|May 28, 2008

Algebraic Geometry and Number Theory18 references3 citations

TL;DR

This paper establishes explicit upper bounds for the bicanonical and pluricanonical maps of $n$-dimensional nonsingular projective varieties of general type by extending Koll{\'a}r's technique. It provides an explicit function $\varepsilon(\iota)$ such that $|mK_V + \lceil Q\rceil|$ is birational for all $m \geq \varepsilon(\iota)$, where $Q$ is a nef $\mathbb{Q}$-divisor and $\iota$ is the dimension of the image of the $m_0$-canonical map. The key contribution is a uniform bound depending on $m_0$ and $n$, especially for $n=4$, yielding $\mu_4 \leq 151m_0 + 77$. The results are derived via induction on fiber dimensions and effective positivity estimates using Riemann-Roch-type inequalities and multiplier ideal techniques.

ABSTRACT

We extend Kollar's technique to look for an explicit function h(n) with phi_m birational onto its image for all integers $m\geq h(n)$ and for all n-dimensional nonsingular projective varieties of general type.

Motivation & Objective

• To find an explicit constant $\mu_n$ such that the $m$-canonical map $\varphi_m$ is birational for all $m \geq \mu_n$ and all $n$-dimensional nonsingular projective varieties of general type.
• To extend Koll{\'a}r's technique for effective birationality of linear systems $|mK_V + \lceil Q\rceil|$ where $Q$ is a nef $\mathbb{Q}$-divisor.
• To provide a uniform bound $\varepsilon(\iota)$ depending on $m_0$ and the dimension $\iota$ of the image of the $m_0$-canonical map, ensuring birationality for $m \geq \varepsilon(\iota)$.
• To address the open problem of explicit birationality bounds in dimension $n \geq 4$, particularly for $n=4$, where a sharp bound is derived.

Proposed method

• The method uses induction on the fiber dimension of the $m_0$-canonical fibration, reducing the problem to lower-dimensional varieties via restriction and vanishing theorems.
• It applies effective positivity estimates for line bundles on fibers, using the inequality $\pi^*(K_X)|_F \geq \gamma \sigma^*(K_{F_0})$ with $\gamma > \frac{1}{2m_0+1}$, though optimality is left as an open question.
• The construction relies on a recursive sequence $w_t$ defined by $w_i = \widetilde{\lambda}_i + w_{i-1}(2\widetilde{\lambda}_i + 1)$, with $w_4 = 151\widetilde{\lambda}_4 + 75$, where $\widetilde{\lambda}_i = m_0$ for the top dimension and $\lambda_i$ otherwise.
• The proof uses multiplier ideal techniques and the base point free theorem to ensure the non-vanishing and birationality of the linear systems $|K_F + \lceil \cdots \rceil|$ at each induction step.
• The key technical tool is the use of the $m_0$-canonical fibration to induce a filtration of the variety by fibers of decreasing dimension, allowing recursive control of the pluricanonical systems.
• The final bound is derived by analyzing the number of steps in the induction and the growth of the sequence $w_t$, leading to explicit expressions in terms of $m_0$, $n$, and $\iota$.

Experimental results

Research questions

• RQ1 What is the smallest integer $\mu_n$ such that the $m$-canonical map is birational for all $m \geq \mu_n$ and all $n$-dimensional varieties of general type, for $n \geq 4$?
• RQ2 Can Koll{\'a}r's method be generalized to control the birationality of $|mK_V + \lceil Q\rceil|$ for arbitrary nef $\mathbb{Q}$-divisors $Q$?
• RQ3 What is the optimal constant $\gamma$ such that $\pi^*(K_X)|_F \geq \gamma \sigma^*(K_{F_0})$ holds on fibers, and how does this affect the final bound?
• RQ4 How can the Riemann-Roch formula for $\chi(\mathcal{O}_V(mK_V))$ be used to improve effective bounds when such a formula is unavailable?

Key findings

• For $n$-dimensional nonsingular projective varieties of general type with $P_{m_0} \geq 2$, the linear system $|mK_V + \lceil Q\rceil|$ is birational for all $m \geq \varepsilon(\iota)$, where $\varepsilon(\iota)$ is explicitly defined based on $\iota = \dim \overline{\varphi_{m_0}(V)}$.
• When $\iota \geq n-2$, the bound is $\varepsilon(\iota) = \min\{4m_0+4, 57\} \cdot (2m_0+1)^{n-3} + m_0(n-2) + 2$, which gives a uniform control depending on $m_0$ and $n$.
• For $\iota = n-3$, the bound is $\varepsilon(\iota) = 75(2m_0+1)^{n-3} + m_0(n-3) + 2$, improving on previous estimates in this case.
• When $\iota \leq n-4$, the bound is $\varepsilon(\iota) = (2m_0+1)^{\iota-1} w_{n-\iota+1} + m_0(\iota-1) + 2$, where $w_t$ is defined recursively with $w_4 = 151\widetilde{\lambda}_4 + 75$ and $\widetilde{\lambda}_i = \lambda_i$ for $i < n-\iota+1$, $\widetilde{\lambda}_{n-\iota+1} = m_0$. This provides the sharpest bound in the most general case.
• For $n=4$, the paper gives a sharp bound: $\varphi_m$ is birational for all $m \geq 151m_0 + 77$, which is a concrete solution to Problem 1.1 in dimension 4.
• The results imply that an explicit $\mu_n$ exists if and only if explicit constants $\rho_k$ exist such that $P_{\rho_k} \geq 2$ for all $k$-dimensional varieties of general type, highlighting the importance of effective plurigenus bounds.