Factorial rational varieties which admit or fail to admit an elliptic $\mathbb{G}_m$-action (1812.04979v1)

Abstract: Over a field $k$, we study rational UFDs of finite transcendence degree $n$ over $k$. We classify such UFDs $B$ when $n=2$, $k$ is algebraically closed, and $B$ admits a positive $\mathbb{Z}$-grading, showing in particular that $B$ is affine over $k$. We also consider the Russell cubic threefold over $\mathbb{C}$, and the Asanuma threefolds over a field of positive characterstic, showing that these threefolds admit no elliptic $\mathbb{G}_m$-action. Finally, we show that, if $X$ is an affine $k$-variety and $X\times\mathbb{A}^m_k\cong_k\mathbb{A}^{n+m}_k$, then $X\cong_k\mathbb{A}^n_k$ if and only if $X$ admits an elliptic $\mathbb{G}_m$-action.

• The paper investigates conditions under which factorial rational varieties (UFDs) admit elliptic {G}_m-actions, using Mori's framework for positively {Z}-graded UFDs.
• Key findings include characterizing two-dimensional rational affine UFDs admitting grading and demonstrating that the Russell cubic and Asanuma threefolds do not admit an elliptic {G}_m-action.
• The study utilizes signature sequences to analyze prime and irreducible elements in rational UFDs with {Z}-grading and discusses implications for affine space cancellation problems in algebraic geometry.

Analyzing Rational UFDs and Elliptic G-Action Varieties

This paper, authored by Gene Freudenburg and Takanori Nagamine, investigates the properties of rational Unique Factorization Domains (UFDs) of finite transcendence degree over a field $k$. The paper primarily covers the conditions under which such UFDs admit elliptic $G_m$-actions, focusing on a certain class of factorial affine varieties.

Key Contributions

The authors follow Mori's framework in the classification of UFDs with a positive $\mathbb{Z}$-grading over $k$, specifically targeting those that allow an elliptic $G_m$-action. A significant portion of the paper is dedicated to the understanding of rational UFDs, predominantly two-dimensional affine $k$-domains, and their classification within certain algebraic structures and properties. The Russell cubic threefold and Asanuma threefolds serve as noteworthy examples in the context of elliptic action discussions.

Main Results

1. Characterization of Rational UFDs: The work successfully characterizes two-dimensional rational affine UFDs, providing explicit conditions under which these domains admit a positive $\mathbb{Z}$-grading. The authors offer complete descriptions in the case of algebraically closed fields, concluding that the set $U_k(2, G, R)$ is equivalent to $U_k(2, A)$ under these conditions.
2. Elliptic Action and Non-Contractibility: Notable findings are presented regarding the Russell cubic and Asanuma threefolds, where these structures do not admit an elliptic $G_m$-action. This is a pivotal insight as it rules out a stronger form of contractibility for these varieties.
3. Signature Sequences and UFD Criteria: The paper extends the use of signature sequences within rational UFDs exhibiting a $\mathbb{Z}$-grading, demonstrating their utility in establishing prime and irreducible elements within these rings.
4. Implications on Algebraic Varieties: The implications of the findings on algebraic geometry are considerable, as they offer insights into the autoreducibility and structure of affine varieties, with conjectures pointing towards the characterization of such varieties as affine spaces under specific dimensional constraints.

Implications and Future Directions

This investigation enhances our understanding of UFDs with elliptic actions and provides a foundational basis for exploring more complex algebraic structures. The results prompt further exploration of affine space cancellation problems, particularly in positive characteristic, suggesting potential areas of advancement in fields admitting various gradings. Moreover, the conjecture posed in the paper about the characteristics of the affine space could lead to new approaches in resolving longstanding questions within algebraic geometry and the theory of locally nilpotent derivations.

Future research may focus on expanding these notions to higher-dimensional varieties and exploring the broader implications of elliptic $G_m$-actions in different geometric and algebraic contexts. Additionally, cross-characteristic analysis could unveil intricate relationships between different algebraic field structures in the presence of annulled elliptic actions.

This paper, therefore, provides a comprehensive examination of the nuanced interactions between UFDs, elliptic action allowances, and the associated geometric implications, with promising avenues for subsequent studies.