Smooth Fano varieties with pseudoindex equal to half of their dimension

TL;DR

This work classifies $n$-dimensional smooth Fano varieties with pseudoindex ${\iota_X}=\frac{n}{2}$ and Picard number ${\rho_X}>1$ that admit a birational contraction of an extremal ray. The author combines Mori theory and the deformation theory of rational curves to reduce the possibilities to blow-ups of ${\mathbb P}^n$ or ${Q^n}$ along suitable centers or to projective-bundle models over low-dimensional bases, with several cases where ${\rm i}_X=1$ and ${\iota_X}=\frac{n}{2}$. In the small-contraction scenario, $X$ is realized as a $\mathbb{P}^{\iota_X-1}$-bundle over ${\mathbb P}^{\iota_X+1}$, yielding a concrete vector-bundle description $X\cong {\mathbb P}( {\mathcal O}_{\mathbb P^{\iota_X+1}}^{\oplus 2}(1)\oplus {\mathcal O}_{\mathbb P^{\iota_X+1}}^{\oplus \iota_X-2})$. In the divisorial-contraction case, $X$ must be one of the listed blow-ups of ${\mathbb P}^n$ or ${Q^n}$ along linear or quadrics, or related center configurations, in line with the AO02 and Wis results. Overall, the paper delivers a complete, explicit birational classification in this borderline regime and contributes to the broader understanding of the generalized Mukai conjecture by pinpointing the exact structures that occur when the pseudoindex equals half the dimension.

Abstract

Let $X$ be a complex smooth Fano variety of dimension $n$. Assume that $X$ admits a birational contraction of an extremal ray. In this paper, we give a classification of such $X$ when the pseudoindex is equal to $\frac{\dim X}{2}$.

Theorems & Definitions (41)

• Conjecture 1.1: BCDD03
• Theorem 1.2: Wis90b
• Theorem 1.3
• Definition 2.1
• Proposition 2.2: Ionescu-Wiśniewski inequality Ion86, Wis91
• Theorem 2.3: HNov13
• Theorem 2.4: AO02
• Lemma 2.5
• proof
• Lemma 2.6: BCDD03