Truncated Grassmannians, blow-ups along Schubert varieties and collineations

Abstract: Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail.In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.

• The paper presents an explicit construction of blow-ups of Grassmannians along P⁻-invariant Schubert varieties using truncated Grassmannians.
• It employs orbit closures of unipotent group actions and Demazure modules to derive birational correspondences and projective embeddings.
• The approach yields smooth desingularizations with fibers described by partial collineation varieties, offering insights into degeneracy loci.

Truncated Grassmannians, Blow-Ups Along Schubert Varieties, and Collineations

Overview

This paper develops a unifying framework to describe blow-ups of complex Grassmannians and more general flag varieties along Schubert varieties, utilizing orbit closures of unipotent group actions in truncated projective spaces, and relates the resulting geometric structures to varieties of partial collineations. The work is anchored in an explicit construction for blow-ups in the Grassmannian context, providing resolutions, embeddings via truncated spaces, and birational correspondences among a rich family of auxiliary varieties. The approach is generalized to arbitrary flag varieties using highest weight representations and Demazure modules.

Truncated Grassmannians and Their Construction

Truncated Grassmannians are introduced as orbit closures of abelian unipotent subgroup actions on degree-truncated projectivizations of wedge powers of a vector space $V$. Concretely, for $Gr_d(V)$, the Grassmannian of $d$-dimensional subspaces in $V$ ($\dim V = n$), with a decomposition $V = V_d \oplus V_{n-d}$, attention is focused on the subgroup $\exp(\mathfrak{n}_d)$ acting on the projective space associated to quotients $(\Lambda^d V)_r$ of the full wedge power $\Lambda^d V$ by vectors having more than $r$ components in $Gr_d(V)$0. The resulting varieties $Gr_d(V)$1 interpolate between the affine space $Gr_d(V)$2 ($Gr_d(V)$3) and the full Grassmannian ($Gr_d(V)$4).

All $Gr_d(V)$5 admit a $Gr_d(V)$6-action, where $Gr_d(V)$7 is the opposite maximal parabolic. These varieties are pairwise birationally isomorphic and are related via explicit birational maps, whose graph closures are essential to the blow-up description.

Blow-Ups Along Schubert Subvarieties

The main result for Grassmannians is an explicit identification of the blow-up of $Gr_d(V)$8 along certain $Gr_d(V)$9-invariant Schubert subvarieties $d$0 (those defined by $d$1) with the closure of the graph of the birational map from $d$2 to $d$3. This closure provides a concrete embedding of the blow-up into the product $d$4, realized as the closure of the $d$5-orbit of $d$6. The fibers of the natural projection from this blow-up to $d$7 are described via varieties of partial collineations, reflecting the degeneracy loci structure of the intersection with the exceptional locus.

A sequence of spectral and representation-theoretic results is employed to explicitly resolve the ideal sheaf of $d$8, compute its global sections, and describe the corresponding geometric quotients.

Mixed Blow-Ups and the Space of Collineations

The paper generalizes to multi-center blow-ups, considering sequences of Schubert subvarieties $d$9 and realizing the corresponding multi-blow-up as a subvariety of a product of Grassmannian and several $V$0 factors. The fibers of these projections are identified with spaces of partial collineations, generalizing the classical complete collineation varieties. The maximal blow-up, along all possible $V$1, is proven to be smooth and serves as a desingularization for any other mixed blow-up in this family.

The explicit connection to spaces of collineations both clarifies the nature of the singularities of the blow-ups and provides structural insights into their resolutions.

Extension to Flag Varieties

A generalization to arbitrary flag varieties $V$2 (especially $V$3) is provided. Here, projective embeddings are obtained via equivariant line bundles associated to dominant weights, and Schubert varieties correspond to Demazure submodules within the representation $V$4. The truncated module $V$5 supplies the ambient space for the truncation analog $V$6, and the blow-up of $V$7 along $V$8 is canonically embedded in $V$9.

The consideration of collections of Weyl group elements, leading to iterated/mixed blow-ups, suggests further links with combinatorial and geometric representation theory. The author conjectures smoothness for the variety associated with a full collection of truncated flags corresponding to all Weyl group elements.

Numerical and Structural Results

• The codimension of $\dim V = n$0 in $\dim V = n$1 is $\dim V = n$2.
• All $\dim V = n$3 and associated blow-ups share a common open affine cell (the big $\dim V = n$4-orbit), yielding birational isomorphism classes.
• Fibers over points of $\dim V = n$5 are explicitly described in terms of partial collineation varieties, with dimensions and singularities directly computable from intersection ranks.

Notably, for $\dim V = n$6, all fibers of the blow-up are projective spaces, a case admitting full explicit description.

Theoretical and Practical Implications

The framework offers a new perspective on the structure of blow-ups of Grassmannians and flag varieties, providing canonical projective models, embeddings, and desingularizations in terms of truncated representations and group orbits. The explicit connections with collineation varieties and determinantal geometry unify determinantal and representation-theoretic approaches to these blow-up constructions.

Practically, these results inform the study of degeneracy loci, birational geometry, and the moduli-theoretic interpretation of orbit closures. The identification of smooth desingularizations via maximal mixed blow-ups has direct implications for calculations in intersection theory, vector bundle geometry, and the theory of moduli spaces of linear subspaces with prescribed incidence.

Theoretically, the connection to Demazure modules, highest weight representations, and projective geometry signals new avenues for bridging combinatorial representation theory and birational algebraic geometry. These constructions may yield alternative descriptions or resolutions for orbit closures or degeneracy loci arising in a broad class of moduli problems.

Future Directions

Potential research directions include:

• Extending the explicit desingularization results to other homogeneous varieties and more general orbits.
• Investigating deeper geometric and cohomological properties of the mixed blow-ups and their connections to moduli problems.
• Studying the degenerations and special fibers of these constructions, especially in relation to equivariant degenerations and PBW theory.
• Testing the conjecture regarding the smoothness of generalized blow-ups associated with all Weyl group elements.
• Exploring applications to quantum and enumerative geometry, leveraging the uniform description of (truncated) flag varieties.

Conclusion

The paper provides a unifying, representation-theoretic, and geometrically explicit framework for the study of blow-ups of Grassmannians and flag varieties along Schubert varieties. By leveraging truncated Grassmannians, group actions, and the geometry of collineations, the work advances the understanding of the birational and singularity structure of such blow-ups, offering new tools for both explicit calculations and abstract geometric investigations.