[Paper Review] Bounding the volumes of singular Fano threefolds

Ching-Jui Lai|arXiv (Cornell University)|Apr 12, 2012

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper establishes an effective upper bound for the anticanonical volume $(-K_X)^3$ of $ ational$-factorial $ ational$-Fano threefolds with Picard number one and $ ational$-klt singularities, under a fixed $ ational$-klt threshold $\epsilon$. Using a combination of log minimal model program techniques, surface volume bounds, and Cartier index estimates, it proves that the volume is bounded by a function of $\epsilon$, with explicit dependence on $M(2,\epsilon)$ and $R(2,\epsilon)$, and shows that volumes grow at most as $O(1/\epsilon^3)$, suggesting the existence of higher-volume examples beyond the standard cone construction.

ABSTRACT

Let $(X,Δ)$ be an $n$-dimensional $ε$-klt log $\QQ$-Fano pair. We give an upper bound for the volume ${ m Vol}(-(K_X+Δ))=(-(K_X+Δ))^n$ when $n=2$ or $n=3$ and $X$ is {$\QQ$-factorial} of $ρ(X)=1$. This bound is essentially sharp for $n=2$. Existence of an upper bound for anticanonical volumes is related the Borisov-Alexeev-Borisov Conjecture which asserts boundedness of the set of $ε$-klt log $\QQ$-Fano varieties of a given dimension $n$.

Motivation & Objective

• To establish an effective upper bound for the anticanonical volume $(-K_X)^3$ of $\mathbb{Q}$-factorial, $\rho(X)=1$ singular Fano threefolds with $\epsilon$-klt singularities.
• To support the Borisov-Alexeev-Borisov (BAB) Conjecture by proving boundedness of volumes under fixed $\epsilon$-klt conditions.
• To investigate whether $\epsilon$-klt Fano threefolds can have volumes growing faster than $O(1/\epsilon^2)$, as seen in standard cone constructions.
• To provide explicit quantitative bounds using surface-level volume and Cartier index estimates.

Proposed method

• Apply the log minimal model program to reduce the problem to studying log del Pezzo surfaces via a generalized surface fibration construction.
• Use the bound $M(2,\epsilon) \leq \max\{64, 16/\epsilon + 4\}$ on the volume of $\epsilon/2$-klt del Pezzo surfaces of $\rho=1$.
• Establish an upper bound $R(2,\epsilon) \leq 2(4/\epsilon)^{128 \cdot 2^5 / \epsilon^5}$ for the Cartier index of $K_S$ on such surfaces.
• Derive a volume bound for threefolds by relating the threefold volume to the surface volume and Cartier index via intersection theory on a resolution.
• Use the cone construction as a benchmark, showing that standard cones yield volumes $O(1/\epsilon^2)$, and compare to the new upper bound.
• Apply intersection-theoretic inequalities involving $-K_{Y'_u}$ and exceptional divisors to derive the final volume bound.

Experimental results

Research questions

• RQ1 Can $\epsilon$-klt $\mathbb{Q}$-Fano threefolds with $\rho(X)=1$ have anticanonical volumes growing faster than $O(1/\epsilon^2)$?
• RQ2 Is there a uniform upper bound for $(-K_X)^3$ depending only on $n=3$ and $\epsilon$ for $\epsilon$-klt $\mathbb{Q}$-Fano threefolds with $\rho(X)=1$?
• RQ3 Does the existence of such a bound support the Borisov-Alexeev-Borisov Conjecture in dimension three?
• RQ4 Can the Cartier index and volume of $\epsilon$-klt del Pezzo surfaces be effectively bounded in terms of $\epsilon$?
• RQ5 What is the sharp asymptotic growth rate of the volume of $\epsilon$-klt Fano threefolds as $\epsilon \to 0$?

Key findings

• The anticanonical volume $(-K_X)^3$ of an $\epsilon$-klt log $\mathbb{Q}$-Fano threefold with $\rho(X)=1$ is bounded above by $\left(\frac{24M(2,\epsilon)R(2,\epsilon)}{\epsilon} + 12\right)^3$, where $M(2,\epsilon)$ and $R(2,\epsilon)$ are bounds on surface volume and Cartier index.
• The surface volume bound satisfies $M(2,\epsilon) \leq \max\{64, 16/\epsilon + 4\}$, which grows as $O(1/\epsilon)$ for small $\epsilon$.
• The Cartier index bound satisfies $R(2,\epsilon) \leq 2(4/\epsilon)^{128 \cdot 2^5 / \epsilon^5}$, which grows faster than any polynomial in $1/\epsilon$.
• The resulting threefold volume bound grows as $O(1/\epsilon^3)$, suggesting that higher-volume examples may exist beyond the standard cone construction.
• The cone construction yields volumes of order $O(1/\epsilon^2)$, so the new bound allows for a strictly larger class of examples.
• The paper shows that the volume bound is effective and depends only on $n=3$ and $\epsilon$, supporting the boundedness of $\epsilon$-klt $\mathbb{Q}$-Fano threefolds with $\rho=1$.