Ruled and rational surfaces and their models
Giacomo Graziani
TL;DR
The work develops a comprehensive framework for modeling rational and ruled surfaces over $\mathbb{C}$ via relatively minimal and minimal models, connecting Castelnuovo’s contraction to a Kodaira-type classification through a new type invariant. It establishes foundational tools—ample/very ample criteria, intersection theory, and numerical divisor properties—and analyzes rank-2 bundles and $\mathbb{P}^{1}$-bundles to describe ruled geometries. The classification uses numerical invariants $(q,p_{g},P_{n})$, nef/big/ample theory, and the type $t(\mathcal{S})$ to distinguish complex surfaces, with consequences for rationality and birational geometry of ruled surfaces. The embedded-surface part specializes to Del Pezzo surfaces and scrolls, providing explicit degrees and embeddings (e.g., $\mathbb{F}_{n}$ as scrolls of degree $2k-n$) and finishing with degree-based classifications of minimal and almost minimal surfaces in projective space. Collectively, the results give a rigorous, embedding-aware atlas of ruled and rational surfaces, clarifying when such surfaces are rational and how they sit inside projective spaces via scroll and Del Pezzo constructions.
Abstract
One of the most powerful ideas in the study and classification of algebraic varieties is the notion of a model: that is, to single out an object, in the appropriate isomorphism class, with nice properties. This survey aims to define and study suitable models of rational and, more generally, ruled surfaces in the smooth complex case, and to use them to study such surfaces.