[Paper Review] Unirationality of del Pezzo surfaces of degree two over finite fields (extended version)

Dino Festi, Ronald van Luijk|arXiv (Cornell University)|Aug 1, 2014

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper proves that every del Pezzo surface of degree two over a finite field is unirational, completing a classification initiated by Manin and extended by Salgado, Testa, and Várilly-Alvarado, who left three exceptional cases unresolved. The authors resolve these by constructing rational curves satisfying specific geometric conditions on the branch locus and using field extension techniques, thereby establishing unirationality for all such surfaces over finite fields.

ABSTRACT

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly-Alvarado, who had proved this for all but three surfaces. Over general fields, we state several sufficient conditions for a del Pezzo surface of degree two to be unirational.

Motivation & Objective

• To resolve the remaining three exceptional isomorphism classes of del Pezzo surfaces of degree two over finite fields that were not covered by prior unirationality results.
• To extend the sufficient conditions for unirationality of del Pezzo surfaces of degree two over general fields of characteristic ≠2.
• To provide a constructive proof of unirationality by explicitly building rational curves satisfying geometric constraints on the surface.
• To investigate whether the existence of certain rational curves with prescribed singularities and intersection properties implies unirationality over arbitrary fields.

Proposed method

• Leverages the anti-canonical morphism π: X → P² of degree 2, which realizes the del Pezzo surface as a double cover of P² branched along a quartic curve B.
• Applies a criterion from Salgado, Testa, and Várilly-Alvarado that reduces unirationality to the existence of a rational curve C ⊂ P² satisfying specific intersection and singularity conditions with B.
• Uses Manin’s construction of rational curves via blow-ups and exceptional curves, adapted to cases where the base point lies on the ramification locus.
• Employs algebraic geometry techniques over finite fields, including the fact that any curve birational to P¹ over a finite field extension is already birational over the base field.
• Constructs explicit rational curves of degree 3 or 4 on the three exceptional surfaces, satisfying the hypotheses of case (2) of Corollary 1.3.
• Analyzes the moduli space of such curves via algebraic conditions on coefficients of defining polynomials, identifying loci where the required geometric properties hold.

Experimental results

Research questions

• RQ1 Do all del Pezzo surfaces of degree two over finite fields admit a rational curve that satisfies the unirationality criterion of Salgado, Testa, and Várilly-Alvarado?
• RQ2 Can the existence of a geometrically integral curve C ⊂ P² of degree d ∈{3,4} with an ordinary singular point of multiplicity d−1 at π(P) and even intersection multiplicity with the branch locus B imply unirationality of the surface?
• RQ3 Is there a rational curve on each of the three exceptional del Pezzo surfaces of degree two over finite fields that satisfies the conditions of case (2) of Corollary 1.3?
• RQ4 Does the locus of quartic curves with a triple point at a point Q ∈ B and even intersection multiplicity with B contain a k-rational point for general X and P?
• RQ5 Can the construction of rational curves via Manin’s method be extended to points on the ramification locus, and under what conditions does it yield unirationality?

Key findings

• Every del Pezzo surface of degree two over a finite field is unirational, resolving the final three exceptional cases left open by Salgado, Testa, and Várilly-Alvarado.
• For the three exceptional surfaces, unirationality is established by constructing rational curves of degree 3 or 4 that satisfy the geometric conditions in case (2) of Corollary 1.3.
• The existence of a rational curve C ⊂ P² with an ordinary triple point at π(P) and even intersection multiplicity with the branch locus B implies that the surface X is unirational over a field extension of degree at most 2.
• Over finite fields, any curve that becomes isomorphic to P¹ over a finite extension is already birational to P¹ over the base field, which simplifies the construction of rational curves.
• The locus of curves satisfying the required geometric conditions is non-empty in general, though degenerate components (e.g., reducible or non-reduced curves) must be excluded to ensure geometric integrality.
• A counterexample over F₃ shows that the answer to Question 1.4 can be negative for specific surfaces, but the general case remains open for arbitrary fields of characteristic ≠2.