Curve neighborhoods and combinatorial property $\mathcal{O}$ for a family of odd symplectic partial flag manifolds

TL;DR

This paper characterizes the structure of curve neighborhoods of Schubert varieties in odd symplectic partial flag manifolds and proves a combinatorial version of Conjecture O, advancing understanding in algebraic geometry.

Contribution

It provides a complete description of irreducible components of curve neighborhoods and establishes a combinatorial proof of Conjecture O for these manifolds.

Findings

Explicit description of irreducible components of degree d curve neighborhoods.

Analysis of the lattice structure of these components.

Proof of a combinatorial version of Conjecture O.

Abstract

Let $E$ be an odd dimensional complex vector space and $\mbox{IF}:=\mbox{IF}(1,2;E)$ be the family of odd symplectic partial flag manifold. In this paper we give a full description of the irreducible components of the degree $d$ curve neighborhood of any Schubert variety of $\mbox{IF}$, study their lattice structure, and prove a combinatorial version of Conjecture $\mathcal{O}.$