Blowups of smooth Fano hypersurfaces, their birational geometry and divisorial stability

TL;DR

The paper analyzes the blowup $Y$ of a smooth $n$-dimensional Fano hypersurface $X\subset\mathbb{P}^{n+1}$ along a smooth $k$-dimensional hypersurface $\Gamma$, providing a complete and constructive description of the nef, movable, and effective cones of $Y$ and proving that $Y$ is a Mori dream space with an explicit Cox ring. It determines the full birational model structure, identifies two Mori contractions (to $X$ and to a specific birational model), and describes how $Y$ can fiber over lower-dimensional bases into Fano, Calabi–Yau, or canonically polarized varieties depending on $3-\deg X+\dim\Gamma$. The paper also gives a Sarkisov-link classification in the case $X=\mathbb{P}^n$, and initiates the study of divisorial stability, proving instability (hence non-K-stability) for several blowups, including lines in $\mathbb{P}^n$ and quadrics, with wider conjectures for higher-dimensional centers. Together, these results illuminate the birational geometry of blowups of Fano hypersurfaces and contribute to the understanding of moduli, stability, and fibrations in this setting.

Abstract

Let $Γ$ be a smooth $k$-dimensional hypersurface in $\mathbb P^{k+1}$ and $X \supset Γ$ a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n\geq 3$ and $k\geq 1$. Let $Y \rightarrow X$ be the blowup of $X$ along $Γ$. We give a constructive proof that $Y$ is a Mori dream space. In particular, we describe its Mori chamber decomposition and the associated birational models of $Y$. We classify for which $X$ and $Γ$ the variety $Y$ is a Fano manifold and we initiate the study of K-stability of $Y$.

Theorems & Definitions (48)

• Theorem : = Theorem \ref{['thm:main']}
• Theorem : = Theorem \ref{['thm: Sarkisovlink']}
• Theorem : = Theorem \ref{['thm:stabilityI']}
• Definition 2.1: HuKeel
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 3.1
• proof