Fano manifolds obtained by blowing up along curves with maximal Picard number

TL;DR

This paper classifies Fano manifolds with maximal Picard number obtained by blowing up along curves, identifying specific geometric configurations that achieve this maximum.

Contribution

It provides a complete characterization of Fano manifolds with maximal Picard number resulting from blow-ups along curves, extending the classification of such manifolds.

Findings

Maximal Picard number is 5 for these Fano manifolds.

Characterization of blow-ups of projective space with specific centers.

Classification of special Fano manifolds with maximal Picard number.

Abstract

The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a quadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.