Communications in Number Theory and Physics

Volume 14 (2020)

Number 3

Bounds for smooth Fano weighted complete intersections

Pages: 511 – 553

DOI: https://dx.doi.org/10.4310/CNTP.2020.v14.n3.a3

Authors

Victor Przyjalkowski (Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia; and HSE University, Russian Federation Laboratory of Mirror Symmetry, NRU HSE, Moscow, Russia)

Constantin Shramov (Steklov Mathematical Institute of the Russian Academy of Sciences, Moscow, Russia; and HSE University, Russian Federation Laboratory of Algebraic Geometry, NRU HSE, Moscow, Russia)

Abstract

We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension $n$ as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed $n+1$. Based on this bound we classify all smooth Fano complete intersections of dimensions $4$ and $5$, and compute their invariants.

Keywords

weighted complete intersections, Fano varieties, bounds, Lagrange multipliers

2010 Mathematics Subject Classification

14J45, 14M10

Full Text (PDF format)

The first-named author is supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No. 14.641.31.0001.

The second-named author is supported by the HSE University Basic Research Program, Russian Academic Excellence Project “5-100”, and by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.

Received 11 October 2019

Accepted 6 February 2020

Published 13 July 2020