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Construction and Applications of Trisections of Low Genus on Del Pezzo Surfaces of Degree One

Julie Desjardins, Vojin Jovanovic

TL;DR

This work develops a method to produce pencils of low-genus trisections on del Pezzo surfaces of degree one by constraining the coefficients of a rational elliptic surface $\mathcal{E}: y^2 = x^3 + f(t)x + g(t)$ so that a chosen trisection gains a triple singularity at a point $Q$. The construction yields genus \le 1 trisections and a one-dimensional family (pencil) of such curves; analyzing their interaction with fibers and associated elliptic curves enables rational points to be generated with potential infinite order, paving the way to Zariski-density results. The authors establish criteria (via Theorem intro-ZD) under which the $L$-rational points of a surface $S$ are Zariski dense, and they demonstrate unirationality for subfamilies (including a DP2 case) that admit low-degree dominant maps. They also provide explicit families, Weierstrass models, and computational code, extending the understanding of rational points on minimal-degree-one DP surfaces and connecting to prior works on low-genus multisections. Overall, the paper provides a concrete, adaptable framework to produce and study low-genus trisections, with significant consequences for rational points and unirationality in this classical geometric setting.

Abstract

Consider a rational elliptic surface over a field $k$ with characteristic $0$ given by $\mathcal{E}: y^2 = x^3 + f(t)x + g(t)$, with $f,g\in k[t]$, $\text{deg}(f) \leq 4$ and $\text{deg}(g) \leq 6$. If all the bad fibres are irreducible, such a surface comes from the blow-up of a del Pezzo surface of degree one. We are interested in studying multisections, curves which intersect each fibre a fixed number of times, specifically, trisections (three times). Many configurations of singularities on a trisection lead to a lower genus. Here, we focus on of several them: by specifying conditions on the coefficients $f,g$ of the surface $\mathcal{E}$, and looking at trisections which pass through a given point three times, we obtain a pencil of cubics on such surfaces. Our construction allows us to prove in several cases the Zariski density of the rational points. This is especially interesting since the results in this regard are partial for del Pezzo surfaces of degree one.

Construction and Applications of Trisections of Low Genus on Del Pezzo Surfaces of Degree One

TL;DR

This work develops a method to produce pencils of low-genus trisections on del Pezzo surfaces of degree one by constraining the coefficients of a rational elliptic surface so that a chosen trisection gains a triple singularity at a point . The construction yields genus \le 1 trisections and a one-dimensional family (pencil) of such curves; analyzing their interaction with fibers and associated elliptic curves enables rational points to be generated with potential infinite order, paving the way to Zariski-density results. The authors establish criteria (via Theorem intro-ZD) under which the -rational points of a surface are Zariski dense, and they demonstrate unirationality for subfamilies (including a DP2 case) that admit low-degree dominant maps. They also provide explicit families, Weierstrass models, and computational code, extending the understanding of rational points on minimal-degree-one DP surfaces and connecting to prior works on low-genus multisections. Overall, the paper provides a concrete, adaptable framework to produce and study low-genus trisections, with significant consequences for rational points and unirationality in this classical geometric setting.

Abstract

Consider a rational elliptic surface over a field with characteristic given by , with , and . If all the bad fibres are irreducible, such a surface comes from the blow-up of a del Pezzo surface of degree one. We are interested in studying multisections, curves which intersect each fibre a fixed number of times, specifically, trisections (three times). Many configurations of singularities on a trisection lead to a lower genus. Here, we focus on of several them: by specifying conditions on the coefficients of the surface , and looking at trisections which pass through a given point three times, we obtain a pencil of cubics on such surfaces. Our construction allows us to prove in several cases the Zariski density of the rational points. This is especially interesting since the results in this regard are partial for del Pezzo surfaces of degree one.

Paper Structure

This paper contains 9 sections, 17 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Select Known Results
  4. Constructing the Pencil
  5. A Rational Point on $\mathcal{T}_R$
  6. Next Steps and Applications
  7. Zariski Density
  8. Unirationality and Genus Zero
  9. Examples

Key Result

Theorem 1.1

Let $S/k$ be a del Pezzo surface of degree one given by the form: in the weighted projective space $\mathbb{P}(2,3,1,1)$, where $F,G\in k[Z,W]$ are homogeneous polynomials of degrees four and six, respectively. Let $f(t) := F(t,1)$, and $g(t) := G(t,1)$, and suppose there exists a field extension $L/k$, along with $a,b,t_0\in L$, such that the following all hold: Then, the $L$-rational point $Q

Figures (3)

  • Figure 1: An example of a trisection $\mathcal{T}_R$ in the family $\{\mathcal{T}_R\}_{R\in\mathcal{E}_{t_1}}$. Fibres containing the relevant points have been highlighted.
  • Figure 2: Table showing which coefficients are free for a surface $\mathcal{E}\in\textbf{Fam}_1$, defined in Definition \ref{['def:famone']}.
  • Figure 3: Table showing which coefficients are free and which affect the surface $\mathcal{E}\in\text{Fam}_2$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Corollary 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • Theorem 1.6
  • Example 1.7
  • Definition 2.1
  • Definition 2.2
  • ...and 25 more