Maximal disjoint Schubert cycles in rational homogeneous varieties

TL;DR

This work develops the invariant of effective good divisibility $\text{e.d.}$ for rational homogeneous varieties of classical type and connects it to Schubert calculus via maximal disjoint pairs. It proves that the complete flag manifold $G/B$ has $\text{e.d.}=\mathsf{h}(\mathcal{D})-1$ in the classical types, with explicit exceptions in $D_n$-variants; it then determines $\text{e.d.}(\mathcal{D}(R))$ for many parabolic quotients, identifying when it equals the Coxeter number $\mathsf{h}(\mathcal{D})$ versus $\mathsf{h}(\mathcal{D})-1$. A central payoff is a general criterion: if $\text{e.d.}(M) > \text{e.d.}(\mathcal{D}(R))$, there are no nonconstant morphisms $M\to \mathcal{D}(R)$, yielding broad nonexistence results for maps between Grassmannians, quadrics, and spinor varieties. The paper also provides detailed, computer-assisted analyses of md-pairs in low-rank cases and includes SageMath scripts to reproduce these computations.

Abstract

In this paper we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.5
• Definition 2.6
• Lemma 2.7