Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case where delta(X)=3. We show that there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case where X has Picard number 5, and to Fano varieties having an elementary divisorial contraction sending a divisor to a curve. In particular we complete the classification of Fano 4-folds with delta(X)=3.

Fano manifolds with Lefschetz defect 3

Cinzia Casagrande;Saverio Andrea Secci
2022-01-01

Abstract

Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case where delta(X)=3. We show that there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case where X has Picard number 5, and to Fano varieties having an elementary divisorial contraction sending a divisor to a curve. In particular we complete the classification of Fano 4-folds with delta(X)=3.
2022
163
625
653
https://arxiv.org/abs/2201.02413
Fano varieties, Lefschetz defect, extremal contractions, fibrations in del Pezzo surfaces
Cinzia Casagrande, Eleonora Anna Romano, Saverio Andrea Secci
File in questo prodotto:
File Dimensione Formato  
CRS2022.pdf

Accesso riservato

Tipo di file: PDF EDITORIALE
Dimensione 692.07 kB
Formato Adobe PDF
692.07 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
1-s2.0-S0021782422001362-main.pdf

Accesso riservato

Descrizione: Corrigendum
Tipo di file: PDF EDITORIALE
Dimensione 180.72 kB
Formato Adobe PDF
180.72 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1843044
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 3
social impact