[Paper Review] Mori dream surfaces associated with curves with one place at infinity

Pinaki Mondal|arXiv (Cornell University)|Dec 8, 2013

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper studies rational surfaces arising from curves with one place at infinity, proving their Cox rings are finitely generated—making them Mori dream spaces—by explicitly constructing generators using approximate roots of polynomials. It fully characterizes the global Zariski and Enriques semigroups at infinity, showing they are isomorphic for equisingular pencils, thus answering a question posed in [CPRL02].

ABSTRACT

We study a class of rational surfaces (considered in [Campillo, Piltant and Reguera, 2005]) associated to curves with one place at infinity and explicitly describe generators of the Cox ring and global sections of line bundles on these surfaces. In particular, we show that their Cox rings are finitely generated, i.e. they are Mori dream spaces. We also compute their "global Zariski semigroups at infinity" (consisting of line bundles which have no base points `at infinity') and "global Enriques semigroups" (generated by closures of curves in C^2). In particular, we show that the global Zariski semigroups at infinity and Enriques semigroups of surfaces corresponding to pencils which are equisingular at infinity are isomorphic, which answers a question of [Campillo, Piltant and Reguera-Lopez, 2002]. We also give an effective algorithm to determine if a (rational) surface `admits systems of numerical curvettes' (these surfaces were also considered in [Campillo, Piltant and Reguera, 2005]).

Motivation & Objective

• To establish that rational surfaces associated with curves having one place at infinity are Mori dream spaces by proving finite generation of their Cox rings.
• To explicitly describe generators of the Cox ring and global sections of line bundles on these surfaces using approximate roots of polynomials.
• To characterize the global Zariski semigroup at infinity and the global Enriques semigroup, particularly in terms of tropical generation by approximate roots.
• To resolve a question from [CPRL02] on whether global Zariski and Enriques semigroups are isomorphic for equisingular pencils at infinity.
• To provide an effective algorithm to determine whether a rational surface admits systems of numerical curvettes, extending results from [CPR05].

Proposed method

• Constructs the Cox ring of the surface $X_V$ via approximate roots of polynomials $f_i = F_i|_{\mathbb{C}^2}$, which are shown to generate the ring explicitly.
• Uses tropical generation—closure under coordinatewise maximum—to describe the global Enriques semigroup $P^{st}(X_V)$, showing that the set of approximate roots tropically generates this semigroup.
• Introduces and analyzes the global Zariski semigroup at infinity $\tilde{P}_\infty(X_V)$, defined as the set of divisors with no base points outside $\mathbb{C}^2$, and expresses it in terms of products of approximate roots.
• Applies Puiseux series and semidegrees to analyze the behavior of curves at infinity, particularly through the generic degree-wise Puiseux series $\tilde{\phi}(x,\xi)$ and its conjugates.
• Employs formal characteristic exponents and polydromy orders to study the structure of singularities at infinity, using $p_j$ and $q_j$ to describe the weighted degrees of Puiseux expansions.
• Uses the $c \star_r \phi$ operation to manipulate conjugate series and compare degrees under semidegrees, crucial for proving the key inequality in Lemma 7.6.

Experimental results

Research questions

• RQ1 Are the Cox rings of rational surfaces associated with curves having one place at infinity finitely generated?
• RQ2 Can the global Zariski semigroup at infinity and the global Enriques semigroup be explicitly described in terms of algebraic invariants of the defining polynomials?
• RQ3 If two pencils are equisingular at infinity, are their global Zariski and Enriques semigroups isomorphic?
• RQ4 What is the role of approximate roots in generating the Cox ring and the Enriques semigroup of such surfaces?
• RQ5 Is there an effective algorithm to determine whether a rational surface admits systems of numerical curvettes?

Key findings

• The Cox ring of $X_V$ for $V \in \mathcal{V}$ is finitely generated, confirming that these surfaces are Mori dream spaces.
• The global Enriques semigroup $P^{st}(X_V)$ is tropically generated by the divisors $D_{ij}$ corresponding to approximate roots of the $f_i = F_i|_{\mathbb{C}^2}$.
• The global Zariski semigroup at infinity $\tilde{P}_\infty(X_V)$ is also described explicitly in terms of products of approximate roots, with a precise characterization given in Theorem 5.7 and Corollary 6.3.
• For equisingular pencils $V_1, V_2 \in \mathcal{V}_1$, the global Zariski semigroups and Enriques semigroups of $X_{V_1}$ and $X_{V_2}$ are isomorphic, answering Question 1.1 from [CPRL02] affirmatively.
• An effective algorithm is provided to determine whether a rational surface admits systems of numerical curvettes, based on the structure of the approximate roots and their tropical generation.
• The proof of Lemma 7.6 establishes a key inequality between semidegrees of resultants, showing that $\delta(g_1)/\deg_y(g_1) \geq \delta(g_2)/\deg_y(g_2)$ when $\epsilon_1 \geq \epsilon_2$, which underpins the main results on semigroups.