Plücker degrees of Quot schemes

Abstract: We study the degree of the Plücker embedding $\varpi$ of the Quot scheme of length $l$ quotients of a locally free sheaf on a smooth projective scheme $\mathrm{S}$ of dimension $d\geqslant 1$. This degree is determined by classes in the Chow ring of the symmetric product $\mathrm{S}<sup>{(l)}$, which are given by the pushforward of the powers of $c_{1}(\mathcal{O}<sup>{[l]})$ with respect to the canonical morphism from the Quot scheme to $\mathrm{S}<sup>{(l)}$. We describe a decomposition of these classes, allowing us to compute the (in a certain sense) leading term of $\mathrm{deg} \ \varpi$. We also obtain a higher-dimensional analogue of a classical result of Schubert.

• The paper presents explicit formulas for the Plücker degree of Quot schemes, linking Segre and tautological classes in a higher-dimensional setting.
• It employs Chow-theoretic methods to decompose tautological classes into a combinatorial summand and geometric correction terms associated with diagonal classes.
• The results generalize Schubert’s classical enumeration, bridging Grassmannian counts with complex moduli spaces and advanced intersection theory.

Plücker Degrees of Quot Schemes: A Technical Exposition

Overview

The paper "Plücker degrees of Quot schemes" (2604.02258) presents a thorough investigation of the degree of the Plücker embedding of Grothendieck's Quot schemes, generalizing classical enumerative results for Grassmannians (Schubert calculus) to higher-dimensional base schemes and more general moduli problems. The author establishes explicit formulas for the Plücker degree in terms of tautological classes and Segre classes, introduces new Chow-theoretic invariants associated with Quot schemes, and elucidates the relationship between these invariants and the geometry of the underlying scheme and vector bundle.

Background and Motivation

Let $E$ be a locally free sheaf of rank $r$ on a smooth, projective, irreducible scheme $S$ of dimension $d$ over an algebraically closed field. Grothendieck's $Quot^l_S(E)$ parametrizes length $l$ quotients of $E$, interpolating between the Grassmannian and Hilbert schemes of points. For an ample line bundle $L$ on $S$, Grothendieck identified a canonical closed embedding of $Quot^l_S(E)$ into a Grassmannian, and the subsequent Plücker embedding into projective space via the determinant bundle. Understanding the degree of this embedding, i.e., the Plücker degree, generalizes problems in classical Schubert calculus and has implications for the geometry of moduli spaces.

For $r$0, $r$1 reduces to a Grassmannian, with the degree computed by Schubert in the 19th century, yielding the $r$2-th Catalan number for $r$3. For $r$4, the structure of $r$5 is poorly understood, often reducible and singular, and classical localization methods become subtle or inapplicable.

Main Results

Explicit Formula for $r$6

For $r$7, the author proves a generalization of Schubert's formula for the degree of the Plücker embedding when $r$8:

$r$9

where $S$0, $S$1 denotes the $S$2th Segre class, and $S$3 is an explicit combination of Segre classes involving Jacobi polynomials. The first term corresponds to the combinatorial "leading term," and the second encodes corrections from the geometry of $S$4 and $S$5.

Chow-Theoretic Description via Symmetric Products

The degree $S$6 is characterized as an intersection number on the symmetric product $S$7, via the canonical morphism $S$8 mapping a quotient to its support cycle. The author introduces a family of tautological classes,

$S$9

where $d$0 is the tautological bundle. These classes generalize Segre classes and determine the Plücker degrees by integration. For $d$1 and $d$2, explicit decompositions are obtained.

Decomposition of the Tautological Classes

A decomposition of $d$3 is established, distinguishing a combinatorially explicit summand (from products of Segre classes via symmetric group symmetrization) and a correction term $d$4 supported on the "diagonal" (non-distinct points) of $d$5:

$d$6

This correction is supported on the complement of the configuration space $d$7 (the locus of $d$8 distinct points), and for degree reasons, vanishes in low degrees.

Higher-Dimensional Schubert-Type Formula

The main formula recovers Schubert's Catalan number for $d$9, and provides a higher-dimensional analogue:

$Quot^l_S(E)$0

This result separates the leading (combinatorial) term and the geometric correction.

Implications and Numerical Results

• Numerical aspect: The explicit Plücker degree formulas generalize well-known Grassmannian/Symmetric function counts to moduli spaces with complex geometry, providing new enumerative invariants for Quot schemes over higher-dimensional $Quot^l_S(E)$1.
• Vanishing of correction terms: For $Quot^l_S(E)$2, the correction $Quot^l_S(E)$3 vanishes, explicitly determining the lower-degree tautological classes. For $Quot^l_S(E)$4, $Quot^l_S(E)$5 is determined up to a constant multiple of the diagonal class.
• Cohomological structure: The tautological and correction classes encode subtle invariants of the Quot scheme and its singularities, with potential links to the study of diagonals and Chow groups of symmetric powers and Hilbert schemes.

Theoretical and Practical Implications

This framework provides a bridge between the combinatorial theory of symmetric functions, classical Schubert calculus, and the intersection theory on singular moduli spaces. By providing a formula for the Plücker degrees in terms of Segre and diagonal classes, the results open pathways for:

• Explicit intersection computations: Relations to Hilbert schemes and moduli of sheaves suggest applications in Donaldson-Thomas theory, enumerative geometry, and the study of tautological rings.
• Chow group theory: The introduced classes $Quot^l_S(E)$6 and their decompositions offer promising new invariants for understanding the intersection theory of symmetric powers, Hilbert schemes, and more general Quot schemes.
• Representation theory and combinatorics: Connections to symmetric group invariants and Jacobi polynomials highlight deep algebraic structures underpinning the geometry.

Future Directions

• Explicit formulas for correction terms: The existence of a general formula for $Quot^l_S(E)$7 in terms of Segre and diagonal classes is conjectured but remains open, with significant potential to clarify the geometric side of the enumerative formulas.
• Generalization to singular schemes: Extending intersection-theoretic techniques such as localization or virtual classes to singular Quot schemes.
• Cohomological and motivic lift: Exploring the implications in the context of cohomology or motives, particularly for surfaces, hyperkähler manifolds, and connections to the theory of tautological rings.
• Applications to moduli of sheaves: Given the relevance to Hilbert schemes and Verlinde-type formulas, ramifications for gauge-theoretic and string-theoretic invariants are expected.

Conclusion

The paper provides an explicit, Chow-theoretic framework for calculating Plücker degrees of Quot schemes, interpolating between combinatorial and geometric intersection theory. The decomposition of integrals, identification of tautological classes, and leading term computations represent substantial advancements in the intersection theory of moduli of sheaves. These results elucidate both the classical and the contemporary structure of Quot schemes, with avenues for further exploration in intersection theory, moduli theory, and algebraic combinatorics.