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Rationality and cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces

In-Kyun Kim, Dae-Won Lee, Masatomo Sawahara

TL;DR

The paper addresses the rationality problem and the existence of $\Bbbk$-cylinders for $\Bbbk$-forms of singular del Pezzo surfaces $S_m^n$ arising from blow-ups of the weighted projective plane. It develops an approach combining orbifold Riemann–Roch, the theory of $\Bbbk$-forms, and cylinder constructions to achieve a complete classification of rationality and cylindricity in terms of the blow-up count $n$, the invariant $\ell_S$, and the presence of $\Bbbk$-points on the unique $(-m)$-curve $Q$ on the minimal resolution. The main results cover the regimes $1\le n\le m+3$, $n=m+4$, and $n=m+5$, with explicit corollaries for odd $m$ and for $S_{2u-1}^{n}$, and rely on embedding $\Bbbk$-forms as weighted complete intersections or hypersurfaces and on constructing cylinders via contractions on the minimal resolution. These findings bridge geometric and arithmetic aspects of non-closed-field settings and advance understanding of cylinders and rationality for singular del Pezzo surfaces with high geometric Picard rank. The work also provides methodological tools for inductive cylinder constructions over $\Bbbk$ through vertical-cylinder criteria and birational reductions to simpler base models.

Abstract

In this paper, we study the rationality and cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces which are blow-ups of the weighted projective planes, where $\Bbbk$ is an arbitrary field of characteristic zero.

Rationality and cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces

TL;DR

The paper addresses the rationality problem and the existence of -cylinders for -forms of singular del Pezzo surfaces arising from blow-ups of the weighted projective plane. It develops an approach combining orbifold Riemann–Roch, the theory of -forms, and cylinder constructions to achieve a complete classification of rationality and cylindricity in terms of the blow-up count , the invariant , and the presence of -points on the unique -curve on the minimal resolution. The main results cover the regimes , , and , with explicit corollaries for odd and for , and rely on embedding -forms as weighted complete intersections or hypersurfaces and on constructing cylinders via contractions on the minimal resolution. These findings bridge geometric and arithmetic aspects of non-closed-field settings and advance understanding of cylinders and rationality for singular del Pezzo surfaces with high geometric Picard rank. The work also provides methodological tools for inductive cylinder constructions over through vertical-cylinder criteria and birational reductions to simpler base models.

Abstract

In this paper, we study the rationality and cylindricity of -forms of singular del Pezzo surfaces which are blow-ups of the weighted projective planes, where is an arbitrary field of characteristic zero.
Paper Structure (6 sections, 30 theorems, 33 equations, 1 figure)

This paper contains 6 sections, 30 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Orbifold Riemann--Roch formula
  4. $\Bbbk$-forms
  5. Cylinders in smooth del Pezzo surfaces
  6. Main results and Proofs

Key Result

Lemma 1.1

DK18 Let $\varphi\colon Y\to X$ be a dominant morphism between normal varieties over $\Bbbk$. Then $Y$ contains a vertical $\mathbb{A}_{\Bbbk}^{n}$-cylinder with respect to the morphism $\varphi$ if and only if the generic fiber of $\varphi$ contains an open subset of the form $\mathbb{A}_{\Bbbk(X)}

Figures (1)

  • Figure 1: Blow-ups of weighted projective plane

Theorems & Definitions (57)

  • Lemma 1.1
  • Theorem 1.2: = Theorems \ref{['thm:intermediate']}, \ref{['thm:m+4']} and \ref{['thm:m+5']}
  • Corollary 1.3: = Corollaries \ref{['cor:interm']}, \ref{['cor:m+4']} and \ref{['cor:m+5']}
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3: cf. Poo17
  • Lemma 2.4
  • proof
  • Lemma 2.5: cf. KSC
  • proof
  • ...and 47 more