On surfaces of high degree with respect to the sectional genus

TL;DR

This work analyzes linearly normal complex projective surfaces in $\mathbf{P}^n$ with degree $d$ and sectional genus $g$ under the key constraint $d\geq 2g-1$, forcing Kodaira dimension $-\infty$. It leverages adjoint systems $|mK_{S'}+H|$ on a minimal desingularization $S'$ to obtain a comprehensive birational classification, distinguishing Veronese surfaces, scrolls, Del Pezzo types, and conic-ruling structures, and extends classical results such as Segre’s theorem to both irrational and rational contexts. The paper provides precise nefness and basepoint criteria for the first adjoint, derives sharp numerical bounds linking $d,g,q,a$ and $n$, and uses simple internal projections and Segre-type extensions to characterize large-degree regimes. In the rational case, it delivers a complete taxonomy of empty or non-empty bi- and tri-adjoint systems, yielding explicit projective models as Veronese re-embeddings of cones or their internal projections and clarifying when adjoint maps are birational.

Abstract

We study and classify linearly normal surfaces in $\mathbf{P}^n$, of degree $d$ and sectional genus $g$, such that $d\geq 2g-1$.

Theorems & Definitions (93)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• Remark 3.4
• proof
• Lemma 3.5: m
• Lemma 3.6: cfm
• Lemma 3.7