Rationally cubic connected manifolds II

TL;DR

This paper classifies certain complex projective varieties with rational curves of degree three, showing they are derived from simpler RCC manifolds via blow-ups, and provides a detailed classification for Fano cases with higher Picard number.

Contribution

It characterizes manifolds with degree three rational curves that are not covered by lines, showing they originate from Picard number one RCC manifolds through blow-ups, and classifies Fano RCC manifolds with Picard number at least three.

Findings

Manifolds are obtained from RCC manifolds of Picard number one by blow-ups.

Classifies all Fano RCC manifolds with Picard number ≥ 3.

Provides structural description of complex projective varieties with degree three rational curves.

Abstract

We study smooth complex projective polarized varieties $(X,H)$ of dimension $ n \ge 2$ which admit a dominating family $V$ of rational curves of $H$-degree $3$, such that two general points of $X$ may be joined by a curve parametrized by $V$ and which do not admit a covering family of lines (i.e. rational curves of $H$-degree one). We prove that such manifolds are obtained from RCC manifolds of Picard number one by blow-ups along smooth centers. If we further assume that $X$ is a Fano manifold, we obtain a stronger result, classifying all Fano RCC manifolds of Picard number $\rho_X \ge 3$