Classifying Smooth Quot Schemes

Abstract: The Quot scheme $$\operatorname{Quot}<sup>{q}(\mathcal{O}_{\mathbb{P}<sup>n}<sup>{r})$$ parametrizes the quotients of the trivial vector bundle of rank $r$ on $n$-dimensional projective space that have Hilbert polynomial $q$ and are flat over a base scheme. We identify numerical conditions on the polynomial $q$ that completely determine when this Quot scheme is smooth and irreducible. Our approach also uncovers further geometric features of the projective scheme $$\operatorname{Quot}^{q} (\mathcal{O}_{\mathbb{P}<sup>n}<sup>{r})$$ including the smoothness of the lexicographic point.

• The paper gives a complete classification: smoothness and irreducibility occur precisely for Hilbert polynomials q(t)=s·C(t+n,n)+pλ(t) satisfying explicit rank and partition conditions.
• The authors prove every nonempty Quot scheme has a smooth lexicographic point and use Borel-fixed-point combinatorics, tangent-space dimensions, and path-connectedness to establish irreducibility or detect singularities.
• The results identify Grassmannians, projective spaces, and products with Hilbert schemes as important smooth families, while exceptional partitions such as (2,1) and (2,2,1) produce sporadic singular Quot schemes.

Overview and main result

The paper by Skjelnes, Smith, and Stillman (2608.14424) gives a complete classification of the smooth, irreducible Quot schemes $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$ parametrizing flat quotients of the trivial rank-$r$ bundle on $n$-dimensional projective space with Hilbert polynomial $q$. The classification is expressed through a family of univariate polynomials attached to integer partitions: for a partition $\lambda = (\lambda_1,\dots,\lambda_e)$, the associated polynomial is

$$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$

The main theorem asserts that $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$ is smooth and irreducible exactly when $q(t) = s\binom{t+n}{n} + p_\lambda(t)$ with $r \geqslant s$, $n \geqslant \lambda_1$, and one of three rank-dependent cases holds:

Case	Condition for smoothness
$r$0	$r$1; or $r$2 with $r$3, $r$4; or $r$5 with $r$6, $r$7; or $r$8 or $r$9
$n$0	$n$1 is smooth (combinatorially enumerated via the earlier classification of smooth Hilbert schemes)
$n$2	$n$3

The $n$4 case is degenerate: the Quot scheme is a single point corresponding to the ambient bundle itself. The $n$5 case reduces to Hilbert-scheme smoothness, since the Quot scheme is locally a product of a projective space and $n$6. The genuinely new content is the $n$7 case, which the authors emphasize is, somewhat surprisingly, less complicated than the $n$8 case: four conditions on the partition suffice, in contrast to the seven conditions required for Hilbert schemes. The classification is uniform over $n$9 because only standard Borel-fixed submodules are needed.

Nonemptiness and the pair $q$0

The analysis rests on a numerical criterion, extending Macaulay's theorem to free modules: $q$1 is nonempty if and only if $q$2 for some $q$3 and partition $q$4 with $q$5 and $q$6. Here $q$7 is the rank of the quotient sheaf, and the lexicographic submodule $q$8 realizes the polynomial. A greedy algorithm decides nonemptiness for any given $q$9; for instance, $\lambda = (\lambda_1,\dots,\lambda_e)$0 (with $\lambda = (\lambda_1,\dots,\lambda_e)$1, $\lambda = (\lambda_1,\dots,\lambda_e)$2, or $\lambda = (\lambda_1,\dots,\lambda_e)$3, $\lambda = (\lambda_1,\dots,\lambda_e)$4) and $\lambda = (\lambda_1,\dots,\lambda_e)$5 yield empty Quot schemes.

A key combinatorial lemma characterizes when $\lambda = (\lambda_1,\dots,\lambda_e)$6 admits a nontrivial decomposition $\lambda = (\lambda_1,\dots,\lambda_e)$7: such a decomposition exists if and only if $\lambda = (\lambda_1,\dots,\lambda_e)$8 has a part of size $\lambda = (\lambda_1,\dots,\lambda_e)$9, with exactly two possibilities when the part $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$0 is unique. This lemma, proved via a recursive "parallel sum identity" algorithm on matrices (interpretable as a game on nonnegative matrices, or via residual flags), controls the number of Borel-fixed points. As a consequence, when $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$1, $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$2, or $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$3, the Quot scheme has a unique saturated Borel-fixed point — a condition that is much easier to verify than the analogous Hilbert-scheme statement.

Smoothness of the lexicographic point

The technical core is the proof that the lexicographic point of every nonempty $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$4 is smooth, generalizing the Reeves–Stillman theorem for Hilbert schemes. The proof computes the tangent-space dimension at the lexicographic point via two formulas: $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$5 equals $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$6 or $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$7 depending on whether all parts of $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$8 equal $$p_\lambda(t) = \sum_{i=1}^{e} \binom{t+\lambda_i-i}{\lambda_i-1}.$$9, while $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$0, where $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$1 is the multiplicity of the part $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$2. The authors then construct a full-dimensional family through the lexicographic point without exhibiting an explicit presentation of the corresponding module — a notable departure from the Hilbert-scheme proof, which relied on explicit generators. For $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$3 the family is built from matrices of homogeneous forms of degree $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$4; for larger $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$5, a copairing argument with global sections of $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$6 extends smoothness from rank $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$7 to rank $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$8.

Combining the unique-Borel-fixed-point criterion with smoothness of the lexicographic point and the path-connectedness of Quot schemes yields smoothness and irreducibility whenever $\Quot^{q}(\mathcal{O}_{\mathbb{P}^n}^{\,r})$9, $q(t) = s\binom{t+n}{n} + p_\lambda(t)$0, or $q(t) = s\binom{t+n}{n} + p_\lambda(t)$1. An explicit Gröbner-basis computation for $q(t) = s\binom{t+n}{n} + p_\lambda(t)$2 with $q(t) = s\binom{t+n}{n} + p_\lambda(t)$3 identifies $q(t) = s\binom{t+n}{n} + p_\lambda(t)$4 with an open dense subset of the lexicographic component, illustrating why avoiding explicit presentations simplifies matters.

Recognizable smooth Quot schemes

Three countable families of smooth Quot schemes are identified with classical parameter spaces:

Polynomial data	Quot scheme
$q(t) = s\binom{t+n}{n} + p_\lambda(t)$5	$q(t) = s\binom{t+n}{n} + p_\lambda(t)$6
$q(t) = s\binom{t+n}{n} + p_\lambda(t)$7, $q(t) = s\binom{t+n}{n} + p_\lambda(t)$8	$q(t) = s\binom{t+n}{n} + p_\lambda(t)$9
$r \geqslant s$0, $r \geqslant s$1	$r \geqslant s$2

The first isomorphism follows from the projection formula and relative Serre vanishing. The second uses a closed immersion $r \geqslant s$3, shown to be an isomorphism when $r \geqslant s$4 has no part equal to $r \geqslant s$5 by dimension comparison. A concrete consequence: the Quot scheme containing the twisted tangent bundle $r \geqslant s$6 is $r \geqslant s$7, recovering the rigidity of the tangent bundle under flat deformations, consistent with Siu's global nondeformability result.

Singular Quot schemes

The converse direction exploits the two Borel-fixed points that exist when $r \geqslant s$8 and $r \geqslant s$9: the lexicographic point $n \geqslant \lambda_1$0 and the point $n \geqslant \lambda_1$1. Comparing tangent-space dimensions via the identity $n \geqslant \lambda_1$2 for the Hilbert-scheme lexicographic components, the paper proves that $n \geqslant \lambda_1$3 is singular whenever $n \geqslant \lambda_1$4, $n \geqslant \lambda_1$5, and $n \geqslant \lambda_1$6 has a part equal to $n \geqslant \lambda_1$7 but $n \geqslant \lambda_1$8. The exceptional partition $n \geqslant \lambda_1$9 is then handled separately: smoothness holds exactly when $r$00 and $r$01, with the $r$02 case requiring an explicit computation of $r$03. For $r$04, a parallel argument shows that the Quot scheme is singular if and only if the Hilbert scheme is.

A striking application concerns cotangent bundles: for $r$05, $r$06, $r$07, $r$08, the Quot scheme containing $r$09 is singular, and Macaulay2 computations suggest the multiplicity $r$10 grows exponentially for $r$11, with $r$12 for $r$13 — indicating singularity for all $r$14, though this remains a conjectural extrapolation from limited data.

Two sporadic singular examples

The partitions $r$15 and $r$16 with $r$17, $r$18 yield singular Quot schemes $r$19 and $r$20 that superficially resemble the Hilbert schemes of two skew lines and twisted cubics: each has two smooth rational irreducible components of unequal dimensions (6 and 5, respectively 9 and 8) meeting in a smooth divisor. The components are exhibited explicitly via parametrized families of presentation matrices whose columns, together with relations, form Gröbner bases, with the lexicographic point obtained as a flat limit. The comparison with Hilbert schemes is instructive and non-obvious: $r$21 is singular while $r$22 is smooth for all $r$23; $r$24 behaves like $r$25 for $r$26 rather than the smooth $r$27; and $r$28 is smooth, behaving like $r$29. The authors attribute these discrepancies to whether the underlying curve configuration can be distributed across the summands of the ambient bundle.

Scope and limitations

The results depend essentially on the ambient bundle being trivial, which forces quotients to be generated in a single degree. The authors state that the methods should extend to $r$30, but the characterization of Borel-fixed points becomes substantially more complicated there. For an arbitrary vector bundle $r$31, or for a base other than projective space (even a Grassmannian or smooth toric variety), the authors state that essentially nothing is known about the geometry of the associated Quot schemes; the present techniques, which rely on Borel-fixed points and lexicographic ideals, do not apply. The exponential-growth claim for the cotangent-bundle partitions is explicitly computational and conjectural. The precise conjectural formula for $r$32 is supported by Macaulay2 evidence but is not proved.

Conclusion

The paper settles the smoothness and irreducibility of $r$33 in full generality via purely combinatorial conditions on the pair $r$34, proves smoothness of the lexicographic point for all nonempty Quot schemes of this type, and identifies the first systematic collections of smooth Quot schemes beyond Hilbert schemes. The classification reveals that Quot schemes and Hilbert schemes diverge in both directions — some Quot schemes are singular where the analogous Hilbert scheme is smooth, and vice versa — and the two sporadic singular examples delineate where the analogy breaks down. The natural boundary of the method, namely the triviality of the ambient bundle and the base being projective space, marks the precise extent of what is currently understood.