[Paper Review] New Examples and Non-examples of Mori Dream Spaces when Blowing up Toric Surfaces

Zhuang He|arXiv (Cornell University)|Mar 2, 2017

Commutative Algebra and Its Applications6 references3 citations

TL;DR

This paper investigates when blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces (MDS), establishing a connection to countably many planar interpolation problems. It provides new examples and non-examples of MDS by generalizing a result of González and Karu, and proves that non-MDS cases arise precisely when certain interpolation problems have solutions, offering a numerical criterion for finitely generated Cox rings.

ABSTRACT

We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori Dream Spaces. For some of these toric surfaces, the question whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of Gonzalez and Karu. We give new examples and non-examples of Mori Dream Spaces among these blow-ups.

Motivation & Objective

• To determine for which weighted projective planes $\mathbb{P}(a,b,c)$ the blow-up at the identity point of the torus is a Mori Dream Space.
• To generalize a non-example criterion of González and Karu to broader classes of toric surfaces.
• To establish a connection between the MDS property of blow-ups and the solvability of countably many planar interpolation problems in $\mathbb{P}^2$.
• To provide new explicit examples and non-examples of Mori Dream Spaces via a numerical criterion based on reduced degree and slope configurations.
• To refine the understanding of when blow-ups of toric surfaces fail to be MDS, particularly in cases where the anticanonical divisor is not big.

Proposed method

• Reduces the question of whether $\operatorname{Bl}_e\mathbb{P}(a,b,c)$ is a Mori Dream Space to the solvability of a family of planar interpolation problems in $\mathbb{P}^2$.
• Applies the theory of Cox rings and finite generation: a variety is a Mori Dream Space iff its Cox ring is finitely generated.
• Uses the concept of reduced degree $d'$ and slope configurations to classify blow-ups, especially for $d' \geq 2$.
• Employs a numerical criterion based on the width $w = cg^2 / ab < 1$ and slope inequalities to detect non-MDS cases.
• Applies a computer-assisted classification for small $d'$, particularly $d' = 5,7,9$, to verify non-examples under specific slope sets $S$ and $T$.
• Generalizes the non-MDS criterion of González and Karu by introducing a conjecture (Conjecture 2.10) for broader families of triples $(a,b,c)$.

Experimental results

Research questions

• RQ1 For which triples $(a,b,c)$ is the blow-up of $\mathbb{P}(a,b,c)$ at the identity point of the torus a Mori Dream Space?
• RQ2 When does the blow-up of a toric surface of Picard number one fail to be a Mori Dream Space, and what conditions characterize such failures?
• RQ3 How are the MDS properties of these blow-ups related to the solvability of countably many planar interpolation problems?
• RQ4 Can a numerical criterion be derived to distinguish between MDS and non-MDS blow-ups based on slope configurations and reduced degree?
• RQ5 What is the role of the reduced degree $d'$ and the width $w$ in determining the finite generation of the Cox ring of the blow-up?

Key findings

• The blow-up $\operatorname{Bl}_e\mathbb{P}(a,b,c)$ is not a Mori Dream Space if and only if a family of countably many planar interpolation problems in $\mathbb{P}^2$ all have solutions.
• New non-examples of Mori Dream Spaces are found for triples with $d' = 5, 7, 9$, specifically: $(a,b,c) = (7, 3, 5)$, $(7, 3, 7)$, and $(7, 3, 9)$, under specific slope sets $S$ and $T$.
• For $d' \geq 2$, the only non-examples under the assumption $d' \cdot s_2 \notin \mathbb{Z}$ are those with $S = \{3\}, T = \{2,4\}, d' = 5$; $S = \{3,5\}, T = \{2,4,6\}, d' = 7$; and $S = \{3,5,7\}, T = \{2,4,6,8\}, d' = 9$.
• The paper confirms that the blow-up is a Mori Dream Space when the reduced degree $d' = 0$ or $d' = 1$, and provides a numerical criterion based on slope inequalities and width $w < 1$.
• The method yields a new criterion (Corollary 2.16) that combines previous results and allows classification of blow-ups of $\mathbb{P}(7,b,c)$ with $b,c \leq 70$ via reduced degree and interpolation solvability.
• The paper provides a computer-assisted verification that no additional non-examples exist beyond the three listed cases for $d' = 5,7,9$ under the given constraints.