Smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two

TL;DR

This paper classifies smooth Fano intrinsic Grassmannians of type (2,n) with Picard number two, providing explicit formulas and analyzing their geometric properties, including their relation to Fujita's freeness conjecture.

Contribution

It introduces the concept of intrinsic Grassmannians, classifies smooth Fano cases of type (2,n) with Picard number two, and derives formulas for their enumeration.

Findings

Complete classification of smooth Fano intrinsic Grassmannians of type (2,n) with Picard number two.

Explicit formula for counting such varieties for any n.

Verification that these varieties satisfy Fujita's freeness conjecture.

Abstract

We introduce the notion of intrinsic Grassmannians which generalizes the well known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Pl\"ucker ideal $I_{d,n}$ of the Grassmannian $\mathrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary $n$. We study their geometry and show that they satisfy Fujita's freeness conjecture.