Inverse curve problems on del Pezzo surfaces

TL;DR

This work classifies and solves arithmetic questions for del Pezzo surfaces by combining blowup-type classification with Galois actions on the Picard group. The authors introduce the intersection invariant $I_X$ and counts $(L_X,C_X)$ to distinguish blowups, and they develop Algorithm 1 and Algorithm 2 to realize all admissible blowup-types and inverse-curve data. They prove comprehensive results on the inverse Galois problem and inverse curve problems across degrees over finite, infinite, and Hilbertian fields, including new characteristic $2$ constructions via Artin–Schreier twists. The findings substantially extend classical results for cubic surfaces, provide explicit finite-field tables, and deliver simultaneous surjectivity results that have strong implications for arithmetic enumerative geometry and the arithmetic of rational curves on del Pezzo surfaces.

Abstract

We classify the number of $k$-rational lines and conic fibrations on del Pezzo surfaces over a field $k$ in terms of relatively minimal surfaces and establish rational curve analogues of the inverse Galois problem for del Pezzo surfaces. We completely solve these problems in all degrees over all global, local and finite fields and provide new solutions of the inverse Galois problem in characteristic 2. Our results generalise well-known theorems on cubic surfaces.

Theorems & Definitions (136)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.7
• Remark 1.8
• Definition 2.1