2007/04/30 by Cinzia Casagrande, Casagrande, C. · 2 citations
Mathematics · #14E30 #14J45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.0704.3912
openalex publication_date 2007/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth complex Fano variety. We define and study 'quasi elementary' contractions of fiber type f: X -> Y. These have the property that rho(X) is at most rho(Y)+rho(F), where rho is the Picard number and F is a general fiber of f. In particular any elementary extremal contraction of fiber type is quasi elementary. We show that if Y has dimension at most 3 and Picard number at least 4, then Y is smooth and Fano; if moreover rho(Y) is at least 6, then X is a product. This yields sharp bounds on rho(X) when dim(X)=4 and X has a quasi elementary contraction, and other applications in higher dimensions.