Smooth Hilbert schemes: their classification and geometry
SSMOOTH HILBERT SCHEMES
THEIR CLASSIFICATION AND GEOMETRY
ROY SKJELNES AND GREGORY G. SMITHA
BSTRACT . Closed subschemes in projective space with a fixed Hilbert polynomial are parametrizedby a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomialthat completely determine when the Hilbert scheme is smooth. We also reinterpret these smoothHilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.
Hilbert schemes are crucial for compactifying families of subschemes and constructing modulispaces. Among these parameter spaces, Hilbert schemes of points on a projective surface areexceptional. Being smooth, they have a wider range of applications including deep results inalgebraic geometry, combinatorics, and representation theory; see [Bea83, Hai01, Gro96, Nak97]. Incontrast, little is known about geometric properties of other Hilbert schemes. Even the geometryof Hilb p ( P m ) , the Hilbert scheme parametrizing closed subschemes in projective m -space P m withHilbert polynomial p , is poorly understood when m (cid:62)
3. Although Hartshorne [Har66] showsthat each Hilb p ( P m ) is path-connected, celebrated insights into these Hilbert schemes typicallyhighlight pathologies. For example, Mumford [Mum62] exhibits an irreducible component inHilb t − ( P ) that is generically non-reduced, Ellia–Hirschowitz–Mezzetti [EHM92] show thatthe number of irreducible components in Hilb c t + c ( P ) is not bounded by a polynomial in c , c ,and Vakil [Vak06] proves that every singularity type appears in some Hilb p ( P ) . As a counterpoint,this article classifies the smooth Hilbert schemes Hilb p ( P m ) and describes their geometry.Our primary theorem uses integer partitions to characterize smooth Hilbert schemes. An integerpartition λ is an r -tuple λ : = ( λ , λ , . . . , λ r ) of positive integers satisfying λ (cid:62) λ (cid:62) · · · (cid:62) λ r (cid:62) Theorem A.
The Hilbert scheme
Hilb p ( P m ) is a smooth variety if and only if there exists an integerpartition λ : = ( λ , λ , . . . , λ r ) such thatp ( t ) = r ∑ i = (cid:18) t + λ i − i λ i − (cid:19) and one of the following seven conditions holds: (1) m = , (2) λ r (cid:62) , (3) λ = ( ) or λ = ( m r − , λ r − , ) = ( r − (cid:122) (cid:125)(cid:124) (cid:123) m , m , . . . , m , λ r − , ) where r (cid:62) and m (cid:62) λ r − (cid:62) , (4) λ = ( m r − s − , λ s + r − s − , ) where r − (cid:62) s (cid:62) and m − (cid:62) λ r − s − (cid:62) , (5) λ = ( m r − s − , s + , ) where r − (cid:62) s (cid:62) , (6) λ = ( m r − , ) where r (cid:62) , (7) λ = ( m + ) or r = . Rewriting the polynomial p in terms of an integer partition λ is equivalent to Hilb p ( P m ) beingnonempty [Har66] and is a small modification of the Gotzmann decomposition [Got78].The integer partitions of the classic smooth Hilbert schemes are easily identified. The integerpartition λ = ( λ ) corresponds to the Grassmannian of ( λ − ) -dimensional planes in P m and Mathematics Subject Classification . 14C05; 14J10, 14M15. a r X i v : . [ m a t h . AG ] A ug R. SKJELNES AND G.G. SMITH the integer partition λ = ( m r ) = ( m , m , . . . , m ) corresponds to the Hilbert scheme parametrizinghypersurfaces of degree r in P m . Less well-known, the Hilbert scheme corresponding to an integerpartition λ : = ( λ , λ , . . . , λ r ) satisfying the inequalities λ > λ > · · · > λ r > P m of linear subspaces having dimensions λ i − (cid:54) i (cid:54) r . When λ r =
1, the partial flag variety is generally not the entire Hilbert scheme. All three of these smoothfamilies are covered by condition (2) and the special case λ = ( ) .All seven conditions in Theorem A correspond to Hilbert schemes that are known to be smooth.Results in Fogarty’s article [Fog68] establish that conditions (1) and (6) guarantee smoothness.Serving as our initial inspiration, Staal’s thesis [Sta20] demonstrates that conditions (2) and (3)correspond to smooth Hilbert schemes. More recently, Ramkumar’s preprint [Ram19] shows thatconditions (4) and (5) are also associated to smooth Hilbert schemes. Under condition (7), theHilbert scheme is just one point. The challenge, for us, lies in proving that this list is exhaustive.The conditions in Theorem A have notable differences. Condition (1) forces the ambient space tobe the projective plane P . The other conditions only regulate the integer partition λ . Condition (2)requires that the smallest part λ r be greater than or equal to 2, whereas conditions (3) – (7) specifyparticular partitions. The singular Hilbert schemes with points corresponding to two skew linesor a twisted cubic curve account for the minor discrepancies between conditions (4) and (5).Condition (6) captures the Hilbert schemes with points corresponding to the union of a hypersurfaceand three points. Our classification shows that smooth Hilbert schemes are quite common whilesimultaneously answering Lin’s question [Lin09].Our other major results elucidate the geometry of smooth Hilbert schemes. These advancementsrely on expanding the classical notion of a residual scheme. To be more precise, let D ⊆ X be a closed immersion in P m where D is a hypersurface. The residual scheme of D ⊆ X is theunique closed subscheme Y ⊂ X such that their defining ideal sheaves on P m satisfy I X = I Y · I D .Geometrically, the scheme X is the union of Y and D . A closed immersion Y ⊂ X in P m is a residualinclusion if there exists a linear subspace Λ in P m containing X and a hypersurface D in Λ such that Y is the residual scheme of D ⊆ X in Λ . A residual flag in P m is a chain ∅ = X e + ⊂ X e ⊂ · · · ⊂ X such that, for all 1 (cid:54) i (cid:54) e , the closed immersion X i + ⊂ X i is a residual inclusion. A residualflag extends a partial flag much like a multiset extends a set: the degree of each hypersurface in aresidual flag is analogous to the multiplicity of each element in a multiset. The parameter spacesrepresenting the flags of residual schemes are projective bundles over partial flag varieties.Our second major contribution proves that a general point on the smooth Hilbert schemessatisfying conditions (2) – (7) corresponds to either a residual flag or the union of a residual flagand a point. For all m (cid:62)
3, this describes the closed subchemes being parametrized by a smoothHilb p ( P m ) . In the first case, we deduce that these smooth Hilbert schemes are projective bundlesover partial flag varieties; see Theorem 4.2. For instance, the points on all Hilbert schemes satisfyingconditions (2) and (3) in Theorem A are flags of residual schemes. In the second case, the smoothHilbert schemes are birational to the product of P m and a projective bundle over a partial flag variety.In other words, we realize the smooth Hilbert schemes Hilb p ( P m ) as suitable generalizations ofpartial flag varieties. MOOTH HILBERT SCHEMES 3
The success in classifying these smooth Hilbert schemes suggests new questions that may betractable. What conditions on the integer partition λ imply that Hilb p ( P m ) is irreducible? How doesone extend this result to Quot schemes or flag Hilbert schemes? What is the right analogue if P m isreplaced with a smooth toric variety, a complete intersection, or a Grassmannian? Computational experience and combinatorial strategy.
Although independent of our proofs,calculations in
Macualay2 [M2] were indispensable in the discovery of the results in this paper.Recoding the Hilbert polynomial as an integer partition gives a novel method of sampling fromamong the nonempty Hilbert schemes. Our computational experiments suggest that the number ofparts in the integer partition λ equal to 1 governs the complexity of the intersection graph for theirreducible components in the associated Hilbert scheme.A standard approach for studying varieties equipped with a group action is to examine theirfixed-points and invariant affine open subsets. The action of the general linear group on projectivespace induces a natural action on the Hilbert scheme. A point in Hilb p ( P m ) is Borel-fixed if, underthis action, the stabilizer of the corresponding closed subscheme contains all upper triangularmatrices. Every nonempty Hilb p ( P m ) has distinguished Borel-fixed point called the lexicographicpoint. Reeves–Stillman [RS97] demonstrate that the lexicographic point is always smooth anddetermine the dimension of the unique irreducible component containing it.Proving smoothness for Hilb p ( P m ) , therefore, reduces to calculating the dimension of the tangentspace at the other Borel-fixed points. Using Macaulay2 [M2], we made a systematic search ofthe Borel-fixed points, for all 3 (cid:54) m (cid:54)
7, exposing conditions (2) – (6). We learn, a posteriori, thatconditions (2) – (5) imply that these Hilbert schemes have at most two Borel-fixed points. Moreover,the relevant fixed-points avoid technicalities arising in positive characteristic, thereby producinguniform results. Seeking a geometric explanation for the integer partition, we analyze the irreduciblecomponents of the saturated ideal defining the lexicographic point. Isolating the key features leadsto residual flags and, ultimately, deeper insights into the geometry of smooth Hilbert schemes.2. R
ESIDUAL FLAGS
In this section, we introduce the notion of a residual flag and show that the scheme parametrizingthese objects is smooth and projective. Throughout, we work over a locally noetherian base scheme S and E denotes a coherent O S -module. We write P ( E ) : = Proj (cid:0)
Sym ( E ) (cid:1) for the projectivization ofthe graded symmetric algebra Sym ( E ) . Grassmannians.
For any S -scheme T , let E T denote the pull-back of E to T . A surjection ofcoherent T -modules E T → F gives a closed immersion P ( F ) ⊆ P ( E T ) = P ( E ) × S T . If F is locally free of constant rank n +
1, then the subscheme P ( F ) is a n -plane in P ( E T ) . The setof T -valued points of the functor Gr (cid:0) n , P ( E ) (cid:1) is the set of the n -planes in P ( E T ) . The S -schemerepresenting this functor is projective; see [EGA71, Proposition 9.8.4]. When E is locally free ofconstant rank m +
1, the map Gr (cid:0) n , P ( E ) (cid:1) → S is smooth of relative dimension ( n + )( m − n ) . R. SKJELNES AND G.G. SMITH
Flag varieties.
Consider an integral e -tuple n : = ( n , n , . . . , n e ) such that n > n > · · · > n e (cid:62) S -scheme T , a flag of type n in P ( E T ) is a chain of closed immersions P ( F e ) ⊂ P ( F e − ) ⊂ · · · ⊂ P ( F ) where each P ( F i ) is a n i -plane in P ( E T ) for all 1 (cid:54) i (cid:54) e . The set of T -valued points of the functorFlag (cid:0) n , P ( E ) (cid:1) is the set of flags of type n in P ( E T ) . The S -scheme representing this functor isprojective; see [EGA71, Proposition 9.9.3]. A flag is a succession of Grassmannians. Hence, when E is locally free of constant rank n +
1, it follows that the map Flag (cid:0) n , P ( E ) (cid:1) → S is smooth ofrelative dimension ∑ ei = ( n i + )( n i − − n i ) . Relative divisors.
A closed subscheme D ⊂ P ( E ) is a relative effective Cartier divisor if it is flatover S and its ideal sheaf is invertible. The divisor D has degree d if, for each geometric pointSpec ( k ) → S , the fibre D × S Spec ( k ) is a hypersurface of degree d in P m ∼ = P ( E ) × S Spec ( k ) ; comparewith [KM85, Corollary 1.1.5.2]. Using the dual sheaf E ∗ : = H om ( E , O S ) , we may parametrize thesedivisors in P ( E ) ; see [Fog68, Proposition 1.2] and [Kol96, Exercise 1.4.1.4]. Lemma 2.1.
Assume that E is a locally free sheaf and let E ∗ be its dual. For all nonnegativeintegers d, the S-scheme P (cid:0) Sym d ( E ∗ ) (cid:1) represents the functor of relative effective Cartier divisorsin P ( E ) having degree d.Proof. Let T be an S -scheme and set F T : = (cid:0) Sym d ( E ∗ ) (cid:1) T = Sym d (cid:0) ( E ∗ ) T (cid:1) . Given a line bundle L on T and a surjection F T → L , we see that L ∗ is an invertible subsheaf of F ∗ T . The ideal sheafgenerated by L ∗ in Sym ( E ∗ ) T is invertible and determines a hypersurface of degree d fiberwise in P ( E T ) . Thus, it defines a relative effective Cartier divisor in P ( E T ) of degree d . (cid:3) Residual scheme.
Consider the closed immersion D ⊆ X in P ( E ) where D is a relative effectiveCartier divisor. Let I D and I X denote the ideal sheaves of the closed subschemes D and X in P ( E ) .The residual scheme to D in X is the closed subscheme Y in P ( E ) defined by the colon ideal sheaf I Y : = ( I X : I D ) = I X · I D − . It follows that I X = I Y · I D and X is the union of the subschemes D and Y ; see [Ful98, Definition 9.2.1] and [Fog68, pp. 512–513]. Definition 2.2.
For any positive integer d , a closed immersion Y ⊂ X in P ( E ) is a d-residualinclusion if there exists a relative effective Cartier divisor D in P ( E ) of degree d such that the closedsubscheme Y is the residual scheme to D in X with respect to P ( E ) . Lemma 2.3.
Let Y ⊂ X in P ( E ) be a d-residual inclusion. The map Y → S is flat if and only if themap X → S is flat.Proof.
The existence of a relative effective Cartier divisor D in P ( E ) such that Y is the residualscheme to D in X yields the short exact sequence0 O Y ( − D ) O X O D , where multiplication by a local equation for D defines the injective map. By hypothesis, the sheaf O D is flat over S . We deduce that O Y and O Y ( − D ) are flat over S if and only if O X is flat over S . (cid:3) MOOTH HILBERT SCHEMES 5
Definition 2.4.
Let ( n , d ) : = ( n , d ) , ( n , d ) , . . . , ( n e , d e ) be a sequence of pairs of positive integerssuch that n > n > · · · > n e >
0. For any S -scheme T , a residual flag of type ( n , d ) in P ( E T ) is achain of closed immersions ∅ = X e + ⊂ X e ⊂ X e − ⊂ · · · ⊂ X in P ( E T ) such that, for all 1 (cid:54) i (cid:54) e , the following properties are satisfied:(i) the scheme X i is flat over T ,(ii) the scheme X i is contained in some n i -plane P ( F i ) ⊆ P ( E T ) , and(iii) the closed immersion X i + ⊂ X i is a d i -residual inclusion in P ( F i ) . Remark 2.5.
For any residual flag, the third property for i = e asserts that closed immersion ∅ = X e + ⊂ X e is a d e -residual inclusion. In other words, the closed subscheme X e is a relativeeffective Cartier divisor of degree d e in some n e -plane P ( F e ) ⊆ P ( E T ) . Remark 2.6.
Residual flags generalize flags of linear subspaces. To be more explicit, assume E islocally free of constant rank m + T be an S -scheme. Given a flag Λ e ⊂ Λ e − ⊂ · · · ⊂ Λ of type n in P ( E T ) where m > n , there exists a flag P ( F e ) ⊂ P ( F e − ) ⊂ · · · ⊂ P ( F ) of type ( n + , n + , . . . , n e + ) in P ( E T ) such that the n i -plane Λ i is a hyperplane in P ( F i ) for all1 (cid:54) i (cid:54) e . Setting X e + : = ∅ , we define X i : = Λ i ∪ X i + by a descending induction. It followsthat X i + ⊂ X i is a 1-residual inclusion for all 1 (cid:54) i (cid:54) e . Thus, the chain of closed immersions ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X is a residual flag of type ( n + , ) , ( n + , ) , . . . , ( n e + , ) in P ( E T ) . Example 2.7.
We illustrate how to recursively construct the defining ideal for the closed subschemesin a residual flag. Let ( n , d ) : = ( , ) , ( , ) and set S = Spec ( k ) where k is a field. The residualflags of type ( n , d ) in P : = Proj ( k [ x , x , x , x ]) are nested pairs X ⊂ X in P such that X is aplanar curve of degree 4 and X is a 2-residual scheme in X . The defining ideal of X has the form I X : = (cid:104) f , f (cid:105) where f is a homogeneous polynomial in k [ x , x , x , x ] of degree 1 and f is a homogeneouspolynomial in k [ x , x , x , x ] of degree d = f . The 2-plane P ( F ) containing X is given by the vanishing of the linear form f . The defining ideal of the closedsubscheme X in P has the form I X : = g · I X = (cid:104) g f , g f (cid:105) where g ∈ k [ x , x , x , x ] is a homogeneous polynomial of degree d =
2. Geometrically, the scheme X is the union of the quadratic hypersurface defined by the vanishing of g and the planar quarticcurve X . For the special configuration in which g = f , the defining ideal of the closed subscheme X is I X = (cid:104) f , f f (cid:105) = (cid:104) f (cid:105) ∩ (cid:104) f , f (cid:105) . (cid:5) Functor of residual flags.
The pullback of a residual scheme is again a residual scheme; see[Fog68, Lemma 1.3]. It follows that residual flags define a contravariant functor from the categoryof S -schemes to the category of sets. Let ( n , d ) : = ( n , d ) , ( n , d ) , . . . , ( n e , d e ) be a sequence ofpairs of positive integers such that n > n > · · · > n e and let E be coherent O S -module. For any S -scheme T , the set of T -valued points of the functor Flag (cid:0) n , d , P ( E ) (cid:1) is defined to be the set ofresidual flags of type ( n , d ) in P ( E T ) . R. SKJELNES AND G.G. SMITH
Lemma 2.8.
Assume that ( n , d ) is a pair of positive integers and E is a coherent sheaf on S. LetF be the universal quotient sheaf on the Grassmannian Gr (cid:0) n , P ( E ) (cid:1) and let F ∗ denote its dual. (i) When d > , the S-scheme P (cid:0) Sym d ( F ∗ ) (cid:1) represents the functor of residual flags of type ( n , d ) in P ( E ) . The structure map of this S-scheme is the composition of the canonical maps P (cid:0) Sym d ( F ∗ ) (cid:1) → Gr (cid:0) n , P ( E ) (cid:1) and Gr (cid:0) n , P ( E ) (cid:1) → S. (ii) When d = , the Grassmannian Gr (cid:0) n − , P ( E ) (cid:1) represents the functor of residual flags oftype ( n , ) in P ( E ) .Proof. Let T be an S -scheme. Remark 2.5 shows that residual flags of type ( n , ) in P ( E T ) are ( n − ) -planes, so part (ii) follows immediately. Assume that d >
1. A T -valued point of P (cid:0) Sym d ( F ∗ ) (cid:1) consists of a line bundle L T on T and a surjection Sym d (cid:0) ( F T ) ∗ (cid:1) → L T together witha T -valued point of Gr (cid:0) n , P ( E ) (cid:1) . Since (cid:0) Sym d ( F ∗ ) (cid:1) T = Sym d (cid:0) ( F T ) ∗ (cid:1) , Lemma 2.1 demonstratesthat L T corresponds to a relative effective Cartier divisors of degree d in the n -plane P ( F T ) . The T -valued point of Gr (cid:0) n , P ( E ) (cid:1) corresponds to the n -plane P ( F T ) ⊆ P ( E T ) . Thus, the S -scheme P (cid:0) Sym d ( F ∗ ) (cid:1) represents the residual flags of type ( n , d ) in P ( E ) . (cid:3) Remark 2.9.
When the base scheme S is the spectrum of a field k and d >
1, the parameter spacefor the residual flags of type ( n , d ) in P m : = Proj ( k [ x , x , . . . , x m ]) is the variety of degree d hypersurfaces in n -planes in P m ; see [Ful98, Example 14.7.12]. Latent planes.
The definition of a residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X of type ( n , d ) in P ( E T ) includes the existence of flag of linear subspaces. For all 1 (cid:54) i (cid:54) e , the scheme X i lies in some n i -plane P ( F i ) ⊆ P ( E T ) . When d e >
1, we refer to the set { P ( F i ) | (cid:54) i (cid:54) e } as the latent planes ofthe residual flag. In the special case d e =
1, the scheme X e is itself a ( n e − ) -plane and the latentplanes are { X e } ∪ { P ( F i ) | (cid:54) i (cid:54) e − } . Lemma 2.10.
Let ( n , d ) be the type of a residual flag. (i) When d e > , there exists a morphism Flag (cid:0) n , d , P ( E ) (cid:1) → Flag (cid:0) n , P ( E ) (cid:1) sending a residualflag to its flag of latent planes. (ii) When d e = , there exists a morphism Flag (cid:0) n , d , P ( E ) (cid:1) → Flag (cid:0) n ◦ , P ( E ) (cid:1) sending a residualflag to its flag of latent planes, where n ◦ : = ( n , n , . . . , n e − , n e − ) .Proof. We need to show that the latent planes are unique and form a flag. Let T be an S -schemeand let ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X be a residual flag of type ( n , d ) in P ( E T ) . Suppose that, forsome 1 (cid:54) i (cid:54) e , the scheme X i is contained in two distinct n i -planes P ( F i ) and P ( F i (cid:48) ) . The closedimmersion X i + ⊂ X i is a d i -residual inclusion in P ( F i ) , so there exists a relative effective Cartierdivisor D i in P ( F i ) such that X i is the union of D i and X i + . Since the codimension of X i in P ( F i ) equals 1, we deduce that X i = D i = P ( F i ) ∩ P ( F i (cid:48) ) . It follows that either the n i -plane containing X i isunique or i = D = X , and d =
1. Thus, each scheme X i is contained in a unique plane havingthe dimension of its corresponding latent plane, so both assertions follow. (cid:3) Representability.
The pivotal result in this section shows that the functor of residual flags isrepresentable. Moreover, it realizes this parameter space as a generalization of a partial flag variety.
MOOTH HILBERT SCHEMES 7
Proposition 2.11.
Assume that ( n , d ) : = ( n , d ) , ( n , d ) , . . . , ( n e , d e ) is the type of a residual flagand E is a coherent sheaf on S. For all (cid:54) i (cid:54) e, let F ∗ i denote the dual of the universal quotientsheaf on the Grassmannian Gr (cid:0) n i , P ( E ) (cid:1) . (i) When d e > , we have the Cartesian square Flag (cid:0) n , d , P ( E ) (cid:1) P (cid:0) Sym d ( F ∗ ) (cid:1) × S P (cid:0) Sym d ( F ∗ ) (cid:1) × S · · · × S P (cid:0) Sym d e ( F ∗ e ) (cid:1) Flag (cid:0) n , P ( E ) (cid:1) Gr (cid:0) n , P ( E ) (cid:1) × S Gr (cid:0) n , P ( E ) (cid:1) × S · · · × S Gr (cid:0) n e , P ( E ) (cid:1) . (ii) When d e = , setting n ◦ : = ( n , n , . . . , n e − , n e − ) gives the Cartesian square Flag (cid:0) n , d , P ( E ) (cid:1) P (cid:0) Sym d ( F ∗ ) (cid:1) × S P (cid:0) Sym d ( F ∗ ) (cid:1) × S · · ·× S P (cid:0) Sym d e − ( F ∗ e − ) (cid:1) × S Gr (cid:0) n e − , P ( E ) (cid:1) Flag (cid:0) n ◦ , P ( E ) (cid:1) Gr (cid:0) n , P ( E ) (cid:1) × S Gr (cid:0) n , P ( E ) (cid:1) × S · · · × S Gr (cid:0) n e − , P ( E ) (cid:1) × S Gr (cid:0) n e − , P ( E ) (cid:1) . In both cases, the functor
Flag (cid:0) n , d , P ( E ) (cid:1) is represented by a projective S-scheme.Proof. The bottom arrows in the diagrams are closed immersions; see [EGA71, Proposition 9.9.3].To prove the projectivity of Flag (cid:0) n , d , P ( E ) (cid:1) , it is enough to show that these squares are Cartesian.Assume that d e >
1. Let T be an S -scheme and let ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X be a residual flagof type ( n , d ) in P ( E T ) . For all 1 (cid:54) i (cid:54) e , there exists a relative effective Cartier divisor D i in P ( F i ) such that X i = D i ∪ X i + . Lemma 2.8 implies that the product of projective bundles P : = P (cid:0) Sym d ( F ∗ ) (cid:1) × S P (cid:0) Sym d ( F ∗ ) (cid:1) × S · · · × S P (cid:0) Sym d e ( F ∗ e ) (cid:1) over G : = Gr (cid:0) n , P ( E ) (cid:1) × S Gr (cid:0) n , P ( E ) (cid:1) × S · · · × S Gr (cid:0) n e , P ( E ) (cid:1) parametrizes e -tuples of relativeeffective Cartier divisors of degree d i contained in some n i -plane in P ( E ) for all 1 (cid:54) i (cid:54) e . Hence, wehave a morphism from Flag (cid:0) n , d , P ( E ) (cid:1) to P sending the residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X tothe e -tuple (cid:0) D ⊂ P ( F ) , D ⊂ P ( F ) , . . . , D e ⊂ P ( F e ) (cid:1) . Combined with the morphism in Lemma 2.10,we obtain a morphism from Flag (cid:0) n , d , P ( E ) (cid:1) to the fibre product Flag (cid:0) n , P ( E ) (cid:1) × G P .We want to exhibit the inverse of this morphism. A T -valued point of this fibre product is aflag P ( F e ) ⊂ P ( F e − ) ⊂ · · · ⊂ P ( F ) of type n in P ( E T ) together with an e -tuple of relative effectiveCartier divisors D i in P ( F i ) for all 1 (cid:54) i (cid:54) e . Setting X e + : = ∅ , we define X i : = D i ∪ X i + for all1 (cid:54) i (cid:54) e by descending induction. By construction, we have a chain ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X of closed immersions in P ( E T ) . Since Lemma 2.3 shows that, for all 1 (cid:54) i (cid:54) e , the scheme X i is flat over T , this chain is a residual flag of type ( n , d ) in P ( E T ) . As the pullback of a residualscheme is again a residual scheme, this construction is also functorial. We conclude that the schemeFlag (cid:0) n , d , P ( E ) (cid:1) and the fibre product Flag (cid:0) n , P ( E ) (cid:1) × G P are isomorphic.The proof for the case d e = X e is a ( n e − ) -plane in P ( E T ) ,we have a natural morphism from Flag (cid:0) n , d , P ( E ) (cid:1) to the product P (cid:0) Sym d ( F ∗ ) (cid:1) × S P (cid:0) Sym d ( F ∗ ) (cid:1) × S · · · × S P (cid:0) Sym d e − ( F ∗ e − ) (cid:1) × S Gr (cid:0) n e , P ( E ) (cid:1) sending the residual flag to the tuple (cid:0) D ⊂ P ( F ) , D ⊂ P ( F ) , . . . , D e − ⊂ P ( F e − ) , X e ⊂ P ( E T ) (cid:1) .Using Lemma 2.10, we obtain a morphism from Flag (cid:0) n , d , P ( E ) (cid:1) to the appropriate fibre product.As above, one exhibits the inverse morphism by defining X i : = D i ∪ X i + for all 1 (cid:54) i (cid:54) e − (cid:3) R. SKJELNES AND G.G. SMITH
Corollary 2.12.
Let ( n , d ) : = ( n , d ) , ( n , d ) , . . . , ( n e , d e ) be the type of a residual flag and let Ebe a locally free sheaf on S of constant rank n + where n (cid:62) n . (i) When d e > , the structure map Flag (cid:0) n , d , P ( E ) (cid:1) → S is smooth of relative dimension e ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) . (ii) When d e = , the structure map Flag (cid:0) n , d , P ( E ) (cid:1) → S is smooth of relative dimension − ( n e − − n e ) + e ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) . Proof.
For all 1 (cid:54) i (cid:54) e , the sheaf Sym d i ( F ∗ i ) on the Grassmannian Gr (cid:0) n i , P ( E ) (cid:1) is locally free ofconstant rank (cid:0) n i + d i d i (cid:1) . Using the relative dimensions of flag varieties and Grassmannians, Proposi-tion 2.11 shows that the map Flag (cid:0) n , d , P ( E ) (cid:1) → S is smooth of the claimed relative dimensions. (cid:3) Question 2.13.
When E is locally free, the fibre product interpretation leads to a presentationfor the Chow ring of Flag (cid:0) n , d , P ( E ) (cid:1) . Specifically, one combines the formula for the Chowring of a projective bundle with the description of the Chow ring of a partial flag variety; see[Ful98, Examples 8.3.4 and 14.7.16]. Moreover, the cycle map on Flag (cid:0) n , d , P ( E ) (cid:1) from its Chowring to its integral cohomology ring is an isomorphism; see [Ful98, Example 19.1.11]. Whichaspects of Schubert calculus on partial flag varieties extend to the parameter space of residual flags? Question 2.14.
For line bundles on P ( E ) , the higher-direct images under the structure map tothe base scheme S are well-understood. Similarly, the Borel–Weil–Bott theorem describes thehigher-direct images of line bundles on a flag variety under the structure map to the base scheme.What is the common refinement for line bundles on the parameter space Flag (cid:0) n , d , P ( E ) (cid:1) ?3. H ILBERT POLYNOMIALS AND RESIDUAL FLAGS
Using the geometry of residual flags we explain the combinatorial formula for the Hilbertpolynomials. We show that the type of a residual flag encodes the Hilbert polynomial of its largestsubscheme. We also prove that every lexicographic ideal (other than the zero ideal and unit ideal)determines a residual flag. Let R : = k [ x , x , . . . , x m ] denote the standard graded polynomial ringover a field k and set P m : = Proj ( R ) . Integer partitions.
We repackage the type of a residual flag as a single integer partition. Given asequence ( n , d ) , ( n , d ) , . . . , ( n e , d e ) of pairs of positive integers such that n > n > · · · > n e > λ : = ( λ , λ , . . . , λ r ) satisfies λ (cid:62) λ (cid:62) · · · (cid:62) λ r (cid:62) λ = ( n , n , . . . , n (cid:124) (cid:123)(cid:122) (cid:125) d -times , n , n , . . . , n (cid:124) (cid:123)(cid:122) (cid:125) d -times , . . . , n e , n e , . . . , n e (cid:124) (cid:123)(cid:122) (cid:125) d e -times ) . The length of λ is r : = d + d + · · · + d e . It can be convenient to use a notation for integer partitionsthat indicates the number of times each integer occurs as a part; see [Mac15, Subsection 1.1]. Theexpression λ = ( . . . , i a i , . . . , a , a ) means that, for all positive integers i , exactly a i of the parts in λ are equal to i . For instance, we have λ = ( n d , n d , . . . , n d e e ) . MOOTH HILBERT SCHEMES 9
To describe Hilbert polynomials, we treat a binomial coefficient with a variable in its numeratoras a polynomial. Specifically, for all integers c , we set (cid:18) tc (cid:19) : = (cid:40) c ! ( t )( t − ) · · · ( t − c + ) if c (cid:62)
00 if c < (cid:0) tc (cid:1) has rational coefficients and degree c . Lemma 3.1.
Let ( n , d ) be the type of a residual flag and let λ : = ( λ , λ , . . . , λ r ) be its associatedinteger partition. For any residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X of type ( n , d ) in P m , the Hilbertpolynomial of the closed scheme X in P m isp ( t ) = r ∑ i = (cid:18) t + λ i − i λ i − (cid:19) . Proof.
We proceed by induction on e . The closed immersion X ⊂ X is a d -residual inclusion insome n -plane contained in P m . Hence, there exists a relative effective Cartier divisor D in this n -plane and a short exact sequence of sheaves(3.1.1) 0 O X ( − D ) O X O D . When e =
1, we have X = ∅ . The closed scheme X is the divisor D , so its Hilbert polynomial is p ( t ) = (cid:18) t + n n (cid:19) − (cid:18) t + n − d n (cid:19) = d ∑ i = (cid:20)(cid:18) t + n − i + n (cid:19) − (cid:18) t + n − in (cid:19)(cid:21) = d ∑ i = (cid:18) t + n − in − (cid:19) . The integer partition associated to the residual flag ∅ ⊂ X is λ = ( n d ) , so the base case holds.Suppose that e >
1. The residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X has e − ( n d , n d , . . . , n d e e ) . The induction hypothesis implies that the Hilbert polynomialof the closed scheme X is ∑ ri = d + (cid:0) t + λ i − i + d λ i − (cid:1) . From the short exact sequence (3.1.1), we deducethat the Hilbert polynomial of the closed scheme X is p ( t ) = r ∑ i = d + (cid:18) t + λ i − i + d λ i − (cid:19) + d ∑ i = (cid:18) t + n j − in j − (cid:19) = r ∑ i = (cid:18) t + λ i − i λ i − (cid:19) . (cid:3) Remark 3.2.
Let X be a closed subscheme in P m having Hilbert polynomial p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) .The dimension of X is λ − X is the number d of parts in λ equal to λ . Oneverifies that the arithmetic genus of X is ( − ) λ − r ∑ i = (cid:18) λ i − i λ i − (cid:19) = r ∑ i = ( − ) λ − λ i (cid:18) i − λ i − (cid:19) . Remarkably, the Gotzmann Regularity Theorem shows that the length r of λ is an upper bound onthe Castelnuovo-Mumford regularity of the saturated ideal defining the closed subscheme X in P m ;see [Got78, Lemma 2.9] or [BH93, Theorem 4.3.2] Lexicographic ideals.
We identify a special residual flag by recognizing the geometric propertiesof a distinguished monomial ideal. The lexicographic order on the monomials in R = k [ x , x , . . . , x m ] is defined by declaring x b x b · · · x b m m > x c x c · · · x c m m whenever the first nonzero entry in the integersequence ( b − c , b − c , . . . , b m − c m ) is positive. A lexicographic ideal I is a monomial ideal in R such that, for all integers j , the homogeneous component of I j is the k -vector space spanned bythe dim k I j largest monomials in lexicographic order.As the cornerstone of our approach, we recount a variant of the Macaulay characterization [Mac27]of the Hilbert functions for homogeneous ideals in polynomial ring R . Recall that the Hilbert function h R / I : Z → N of a homogeneous ideal I in R is defined, for all integers j , by h R / I ( j ) : = dim k ( R / I ) j . Lemma 3.3.
Let p be a numerical polynomial having degree less than m. The following statementsare equivalent. (a)
There exists a closed subscheme X in P m whose Hilbert polynomial is p. (b) There exists an integer partition λ : = ( λ , λ , . . . , λ r ) such that p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . (c) There exists an integer partition λ : = ( λ , λ , . . . , λ r ) such that p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) , anda lexicographic ideal L ( λ ) in R such that h R / L ( λ ) ( j ) = ∑ ri = (cid:0) j + λ i − ij − i + (cid:1) for all integers j.Outline of Proof. We sketch the details because a proof may be derived from other accounts ofMacaulay’s work via elementary identities for binomial coefficients; see [GKP94, Table 174]. (a ⇒ b) Let (cid:96) be a fixed sufficiently large integer. One uses the (cid:96) -th Macaulay representation forthe integer p ( (cid:96) ) to obtain an integer partition λ : = ( λ , λ , . . . , λ r ) such that p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) ;see [BH93, Lemma 4.2.6 and Corollary 4.2.14] or [Sta20, Proposition 2.3]. (b ⇒ c) One verifies that the function h : Z → N defined, for all integers j , by h ( j ) : = ∑ ri = (cid:0) j + λ i − ij − i + (cid:1) satisfies the Macaulay inequality (cid:0) h ( j ) (cid:1) (cid:104) j (cid:105) (cid:62) h ( j ) ; see [BH93, Theorem 4.2.10]. (c ⇒ a) One takes X to be the closed subscheme of P n defined by the lexicographic ideal L ( λ ) . (cid:3) Remark 3.4.
By using the conjugate integer partition, Corollary 5.7 in [Har66] gives an alternativecondition equivalent to Lemma 3.3 (b); see [Sta20, Lemma 2.4].We specify monomial generators and a primary decomposition of the lexicographic ideal L ( λ ) ;these generators are also listed in [RS97]. Proposition 3.5.
Let λ : = ( λ , λ , . . . , λ r ) be an integer partition and let a j be the number of partsin λ equal to j, for all positive integers j. When n (cid:62) λ , the corresponding lexicographic ideal isL ( λ ) = (cid:10) x a m + , x a m x a m − + , . . . , x a m x a m − · · · x a m − x a + m − , x a m x a m − · · · x a m − x a m − (cid:11) . Moreover, the unique irredundant irreducible decomposition of this monomial ideal isL ( λ ) = (cid:92) (cid:54) i (cid:54) ma i (cid:54) = (cid:10) x a m + , x a m − + , . . . , x a i + + m − i − , x a i m − i (cid:11) . Proof.
We first establish the decomposition. As each intersectand is generated by powers of thevariables and no two have the same dimension, this intersection is the irredundant irreducibledecomposition of some monomial ideal. It remains to show that this ideal is L ( λ ) . Since a j = j > λ , each irreducible ideal contains (cid:104) x , x , . . . , x m − λ − (cid:105) and we may assume that λ = m . MOOTH HILBERT SCHEMES 11
We proceed by induction on the number e of positive entries in ( a , a , . . . , a m ) . When e =
1, theinteger partition is ( m a m ) and the ideal is (cid:104) x a m (cid:105) . The principal ideal (cid:104) x a m (cid:105) is lexicographic. Since themonomials { , x , . . . , x a m − } form a basis as free k [ x , x , . . . , x m ] -module for the quotient R / (cid:104) x a n (cid:105) ,it follows that h R / (cid:104) x am (cid:105) ( j ) = ∑ a m i = (cid:0) j + m − ij − i + (cid:1) . By Lemma 3.3 (c), we deduce that L ( m a m ) = (cid:104) x a m (cid:105) andthe base case holds.Now, assume that e >
1. Set I : = L ( m a m ) = (cid:104) x a m (cid:105) . The induction hypothesis implies that J : = (cid:92) (cid:54) i (cid:54) m − a i (cid:54) = (cid:10) x a m + , x a m − + , . . . , x a i + + m − i − , x a i m − i (cid:11) is the lexicographic ideal associated to the integer partition ν : = ( λ a m + , λ a m + , . . . , λ r ) . Since theintersection of lexicographic ideals is again lexicographic, it is enough to prove that the ideals I ∩ J and L ( λ ) have the same Hilbert function. From the short exact sequences of graded R -modules0 RI ∩ J RI ⊕ RJ RI + J R ( − a n ) J RJ RI + J , we see that h R / I ∩ J ( j ) = h R / I ( j ) + h R / J ( j ) − h R / ( I + J ) ( j ) = h R / I ( j ) + h R / J ( j − a n ) for all integer j .The equality J = L ( ν ) implies that h R / J ( j ) = ∑ ri = a m + (cid:0) j + λ i − i + a m j − i + a m + (cid:1) , so we deduce that h R / I ∩ J ( j ) = a m ∑ i = (cid:18) j + λ i − ij − i + (cid:19) + r ∑ i = a m + (cid:18) j + λ i − ij − i + (cid:19) = r ∑ i = (cid:18) j + λ i − ij − i + (cid:19) , which by Lemma 3.3 (c) completes the induction.Lastly, we establish that the given set of monomials generate L ( λ ) . The intersection of monomialideals is generated by the least common multiples of their monomial generators, so we observe x a m x a m − · · · x a i + m − i − x a i + m − i = lcm (cid:0) x a i + m − i , x a i + m − i , . . . , x a i + m − i , x a i m − i , x a i + m − i − , . . . , x a m (cid:1) for all 2 (cid:54) i (cid:54) m , x a m x a m − · · · x a m − x a m − = lcm (cid:0) x a m − , x a m − , . . . , x a m (cid:1) . Hence, each of the given monomials is a least common multiple of a generator from the irreduciblecomponents. Since each variable x i appears as a minimal generator in an irreducible componentwith exponent either a m − i or a m − +
1, we see that the least common multiple of any subset ofgenerators for the irreducible components is divisible by at least one of the given monomials, so theopposite inclusion also holds. (cid:3)
We complete our converse to Lemma 3.1 by relating lexicographic ideals to residual flags. Thisgeometric interpretation for the lexicographic ideal L ( λ ) appears to be new. Corollary 3.6.
Let ( n , d ) be the type of a residual flag in P m . For each (cid:54) i (cid:54) e, let X i be theclosed subscheme in P m defined by the lexicographic ideal L ( n d i i , n d i + i + , . . . , n d e e ) . The chain of closedimmersions ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X forms a residual flag of type ( n , d ) in P m .Proof. For all 1 (cid:54) i (cid:54) e , the irreducible decomposition in Proposition 3.5 implies that X i + ⊂ X i and the monomial generators in Proposition 3.5 establish that the closed subscheme X i is containedin the n i -plane in P m defined by the monomial ideal (cid:104) x , x , . . . , x m − n i − (cid:105) . For all positive integers j ,let a j denote the number of parts in the integer partition ( n d , n d , . . . , n d e e ) equal to j . Restricting to the linear subspace P n i : = Proj ( k [ x m − n i , x m − n i + , . . . , x m ]) in P m , Proposition 3.5 also shows that theclosed subscheme X i is defined by the monomial ideal I i : = (cid:10) x a ni + m − n i , x a ni m − n i x a ni + + m − n i − , . . . , x a ni m − n i x a ni + + m − n i − · · · x a m − x a + m − , x a ni m − n i x a ni + + m − n i − · · · x a m − x a m − (cid:11) . It follows that I i = x a ni m − n i · J where J : = (cid:10) x m − n i , x a ni + + m − n i − , . . . , x a ni + + m − n i − · · · x a m − x a + m − , x a ni + + m − n i − · · · x a m − x a m − (cid:11) = (cid:10) x n − n i , x n − n i − , . . . , x n − n i + − (cid:11) + I i + . Since a n i = d i , the closed immersion X i + ⊂ X i is a d i -residual inclusion in P n i . Therefore, the chainof closed immersions ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X is a residual flag of type ( n , d ) in P m . (cid:3) Remark 3.7.
By definition, the lexicographic point in a Hilbert scheme corresponds to the closedsubscheme in P m defined by the lexicographic ideal. Theorem 1.4 in [RS97] proves that thelexicographic point is smooth, so this point lies on a unique irreducible component called thelexicographic component. Theorem 4.1 in [RS97] computes the dimension of this component. When ( n , d ) : = ( n , d ) , ( n , d ) , . . . , ( n e , d e ) is the type of a residual flag, n : = m , and λ is its associatedinteger partition, the lexicographic component determined by L ( λ ) in R : = k [ x , x , . . . , x m ] is (3.7.2) e ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) if n e > d e > − ( n e − − n e ) + e ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) if n e > d e = n d e + e − ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) if n e = d e − > n d e − ( n e − − n e − ) + e − ∑ i = (cid:20)(cid:18) n i + d i d i (cid:19) − + ( n i + )( n i − − n i ) (cid:21) if n e = d e − = n d e if n e = e = Question 3.8.
The saturated monomial ideals defining residual flags of type ( n , d ) in P m determinethe torus-fixed points in the parameter space Flag (cid:0) n , d , P ( E ) (cid:1) . Following Proposition 3.5, theirredundant irreducible decomposition for each such monomial ideal has a combinatorial description.Can one use this perspective to count these monomial ideals and, thereby, compute the Eulercharacteristic of the projective scheme Flag (cid:0) n , d , P ( E ) (cid:1) ?4. G EOMETRY OF SMOOTH H ILBERT SCHEMES
In this section, we identify the Hilbert schemes isomorphic to a parameter space of residual flags.Exploiting this identification, we describe the closed subscheme corresponding to a general point onany smooth Hilbert scheme. Thus, we obtain a birational description of all smooth Hilbert schemes.
Geometry.
Let E be a locally free sheaf on a locally noetherian scheme S . The projective bundle P ( E ) carries a tautological invertible sheaf that is relatively ample over S . We compute the Hilbertpolynomials for closed subschemes in P ( E ) relative to this tautological bundle. For any numericalpolynomial p , the set of T -valued points of the functor Hilb p (cid:0) P ( E ) (cid:1) is the set of closed subschemes X ⊆ P ( E T ) that are flat over the S -scheme T and have Hilbert polynomial p . The S -schemerepresenting this functor is projective; see [Kol96, Theorem 1.4]. MOOTH HILBERT SCHEMES 13
Lemma 4.1.
Let E be locally free sheaf on S of constant rank m + and fix a polynomial p ∈ Q [ t ] .The Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) is smooth over S if and only if the fibre Hilb p ( P m ) is nonsingularover every geometric point of S.Proof. Let X ⊆ P m be a closed subscheme in the fibre of P ( E ) over a geometric point in S . When X corresponds to a smooth point on Hilb p ( P m ) , Theorem 2.10 in [Kol96] proves that the structuremap Hilb p (cid:0) P ( E ) (cid:1) → S is flat at this geometric point. Therefore, this structure map is smooth if andonly if its the fibre is nonsingular over every geometric point. (cid:3) Theorem 4.2.
Let ( n , d ) be the type of a residual flag and let λ : = ( λ , λ , . . . , λ r ) be its associatedinteger partition. Set p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . Assume that E is a locally free sheaf on X of constantrank m + . The natural morphism π : Flag (cid:0) n , d , P ( E ) (cid:1) → Hilb p (cid:0) P ( E ) (cid:1) sending a residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X to the closed subcheme X ⊂ P ( E ) , is anisomorphism if and only if one of the two conditions holds: (2) m (cid:62) λ and λ r (cid:62) , (3) λ = ( ) or λ = ( m r − , λ r − , ) where r (cid:62) and m (cid:62) λ r − (cid:62) .In both cases, the Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) is smooth over S.Proof. Suppose that S = Spec ( k ) for some algebraically closed field k . Theorem 1.1 in [Sta20]demonstrates that conditions (2) and (3) characterize when a nontrivial Hilbert scheme Hilb p ( P m ) has a unique Borel-fixed point. Moreover, Lemma 5.6 in [Sta20] proves that the target Hilb p ( P m ) is nonsingular and irreducible in this situation. Proposition 2.11 and Corollary 2.12 show that thesource Flag (cid:0) κ , d , P ( E ) (cid:1) is a smooth projective variety. Since π is injective, it is enough to certifythat the dimensions of the source and target agree. Using Corollary 2.12 and equation (3.7.2), oneverifies that the dimension of the lexicographic component in Hilb p ( P m ) equals the dimension ofthe parameter space of residual flags if and only if conditions (2) or (3) holds.Suppose that S is any locally noetherian scheme. Lemma 4.1 implies that the Hilbert scheme issmooth. Hence, the source and target of the morphism π are smooth. Since the induced morphismon fibres over any geometric point is an isomorphism, the result follows. (cid:3) Example 4.3.
The conditions in Theorem 4.2 cover the well-known cases of hypersurfaces andGrassmannians. Consider an integer partition λ = ( λ r ) and set p ( t ) : = ∑ ri = (cid:0) t + λ − i λ − (cid:1) . When λ = m ,each point in Hilb p (cid:0) P ( E ) (cid:1) corresponds to a hypersurface of degree r in P ( E ) ; see Lemma 2.8.More generally, each point in Hilb p (cid:0) P ( E ) (cid:1) corresponds to a hypersurface of degree r lying some λ -dimensional linear subspace of P ( E ) . In the special case r =
1, the Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) is the Grassmannian parametrizing ( λ − ) -dimensional linear subspaces in P ( E ) . (cid:5) Example 4.4.
For the integer partition λ = ( ) , Theorem 4.2 shows that each point in the Hilbertscheme Hilb (cid:0) P ( E ) (cid:1) correspond to a hypersurface of degree 2 lying on some line in P ( E ) . Alterna-tively, the Hilbert scheme of two points in P ( E ) is also known to be the blow-up of the diagonal ofthe quotient scheme P ( E ) × S P ( E ) / S , where the symmetric group S on two elements acts bypermuting the factors in the product P ( E ) × S P ( E ) . (cid:5) Example 4.5.
As a novel observation, we identify the Hilbert schemes that are the partial flagvarieties. Consider an integer partition λ : = ( λ , λ , . . . , λ r ) such that m (cid:62) λ > λ > · · · > λ r and set p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . When the integer partition λ also satisfies condition (2) or (3), Theorem 4.2establishes that points in the Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) corresponds to partial flags of type ( λ − , λ − , . . . , λ r − ) in P ( E ) . (cid:5) Remark 4.6.
The geometry of residual flags also explains why the morphism π cannot be surjectivewhen conditions (2) and (3) fail to holds. To avoid these conditions, we may assume that λ r = r (cid:62)
3, and m > λ r − . The smallest scheme X e in the residual flag is a degree d e hypersurface ina line. Hence, the map π cannot be surjective when d e (cid:62)
3. When d e (cid:54)
2, there exists a line Λ e containing X e . The defining properties of a residual flag require the line Λ e to be contained inthe latent plane Λ e − which by assumption has dimension less than m . Since a general line is notcontained such a plane, the map π also not surjective in this case. Remark 4.7.
The strategy outlined in Question 2.13 also leads to a description of the Chow ring(and integral cohomology ring) of the Hilbert schemes classified in Theorem 4.2.
Example 4.8.
Two trivial Hilbert schemes, not covered by Theorem 4.2, are nevertheless particularGrassmannians. When λ = ( m + ) and p ( t ) = (cid:0) t + mm (cid:1) , the Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) is a onepoint corresponding to closed subscheme P ( E ) itself. When r =
0, the Hilbert scheme Hilb (cid:0) P ( E ) (cid:1) is a one point corresponding to empty scheme in P ( E ) . (cid:5) Birationality.
Before examining the birational geometry of the other smooth Hilbert schemes, weremember that some Hilbert schemes split into a product. Let λ = ( m s , λ s + , λ s + , . . . , λ r ) be aninteger partition with m > λ s + and set p ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . We see that p ( t ) = q ( t ) + q ( t − s ) where q ( t ) : = ∑ si = (cid:0) t + m − im − (cid:1) is the Hilbert polynomial for a hypersurface of degree s in P ( E ) and q ( t ) : = ∑ r − si = (cid:0) t + λ i + s − i λ i + s − (cid:1) . Lemma 2.1 shows that the Hilbert scheme parametrizing of thesehypersurfaces is P (cid:0) Sym s ( E ∗ ) (cid:1) and Remark 2 in [Fog68, p. 514] yields there is a natural splitting(4.8.3) Hilb p (cid:0) P ( E ) (cid:1) ∼ = P (cid:0) Sym s ( E ∗ ) (cid:1) × S Hilb q (cid:0) P ( E ) (cid:1) . Given an integer partition λ : = ( λ , λ , . . . , λ r ) , the new integer partition λ ∪ ( ) is defined to be ( λ , λ , . . . , λ r , ) and has length r +
1; see [Mac15, Subsection 1.1].
Proposition 4.9.
Let ( n , d ) be a residual type and λ : = ( λ , λ , . . . , λ r ) be its integer partition. Setp ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . Assume that one of the following conditions holds: (4) λ ∪ ( ) = ( m r − s − , λ s + r − s − , ) where r − (cid:62) s (cid:62) , and m − (cid:62) λ r − s − (cid:62) , or (5) λ ∪ ( ) = ( m r − s − , s + , ) where r − (cid:62) s (cid:62) .The Hilbert scheme Hilb p + (cid:0) P ( E ) (cid:1) is smooth over S. Moreover, a general point on this Hilbertscheme corresponds to the disjoint union of a residual flag of type ( n , d ) and a point.Proof. From the splitting in (4.8.3), it is enough to consider an integer partition λ = ( λ r ) where r (cid:62) λ (cid:62)
2, excluding the two integer partitions ( ) and ( ) . By Lemma 4.1, we manyassume that the base scheme is the spectrum of an algebraically closed field. Over a field ofcharacteristic zero, the Theorem in [Ram19, §0] classifies all Hilbert schemes with precisely MOOTH HILBERT SCHEMES 15 two Borel-fixed points; Theorem 1.1 in [Sta19] shows that this classification also holds over anyalgebraically closed field. Conditions (4) and (5) guarantee that the Hilbert scheme Hilb p + ( P n ) hastwo Borel-fixed points. By computing the dimension of the tangent space at the non-lexicographicBorel-fixed point, Theorem 3.8 in [Ram19] demonstrates that Hilb p + ( P n ) is nonsingular.To understand a general point, consider the universal flag X of type ( λ , s ) in P ( E ) and theuniversal closed subscheme Z having length one on P ( E ) = Hilb (cid:0) P ( E ) (cid:1) . As the structure map isproper, their intersection determines a closed subset in the product Flag (cid:0) λ , s , P ( E ) (cid:1) × S P ( E ) . Let U denote the open complement. There is a morphism ψ : U → Hilb p + (cid:0) P ( E ) (cid:1) induced by sendingthe pair ( X , Z ) to their disjoint union. The source and target of ψ are smooth S -schemes and, usingCorollary 2.12 and equation (3.7.2), one verifies that they have the same relative dimension. Overeach geometric point in S , the induced morphism on the fibres is an open immersion. We concludethat ψ : Flag (cid:0) λ , s , P ( E ) (cid:1) × S P ( E ) (cid:57)(cid:57)(cid:75) Hilb p + (cid:0) P ( E ) (cid:1) is birational map. (cid:3) Remark 4.10.
Under the hypothesis of Proposition 4.9, the Hilbert scheme Hilb p + ( P ( E )) andthe product Flag (cid:0) m , d , P ( E ) (cid:1) × S P ( E ) are birational. However, these schemes are not isomorphic.For instance, the existence of two different Borel-fixed points on the Hilbert scheme implies thatthere is more than one way to embedded a point of multiplicity 1 into the lexicographic ideal; seeProposition 3.5 and Proposition 5.3. Example 4.11.
The integer partition λ = ( r ) is associated the contant Hilbert polynomial r . TheHilbert scheme Hilb r (cid:0) P ( E ) (cid:1) is known to be smooth in two cases: Theorem 2.4 in [Fog68] applieswhen m = r (cid:54)
3. In either case, this Hilbert schemeis birational to the r -fold symmetric product P ( E ) × S P ( E ) × S · · · × S P ( E ) / S r where the symmetricgroup S r on r elements acts by permuting the factors in the product P ( E ) × S P ( E ) × S · · · × S P ( E ) .Using the splitting (4.8.3), this analysis extends to the integer partition λ = ( m r − s , s ) where r (cid:62) s (cid:62)
0. Set p ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . Assuming m = r (cid:54)
3, a general point on Hilb p (cid:0) P ( E ) (cid:1) corresponds to the disjoint union of a hypersurface of degree r − s and s isolated points. (cid:5) Remark 4.12.
The seven conditions in Theorem A imply that the Hilbert scheme Hilb p (cid:0) P ( E ) (cid:1) issmooth over S : Example 4.11 handles conditions (1) and (6), Theorem 4.2 handles conditions (2)and (3), Proposition 4.9 handles conditions (4) and (5), and Example 4.8 handles condition (7). Inparticular, we have a birational description for all of these smooth Hilbert schemes.5. G ENERAL CLASSIFICATION
The final section completes our classification of smooth Hilbert schemes. By identifying enoughsingular points on Hilbert schemes, we prove that our list of smooth Hilbert schemes is exhaustive.
Nearly lexicographic points.
We begin specify a point on a Hilbert scheme by perturbing alexicographic ideal. Geometrically, this nearly lexicographic point corresponds to a residualflag with an embedded point whose nilpotent elements do not lie in the smallest linear subspacecontaining the residual flag.
Lemma 5.1.
Let λ : = ( λ , λ , . . . , λ r ) be an integer partition and set p ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . Fixm > λ and consider both the lexicographic ideal L ( λ ) and the monomial idealJ : = (cid:10) x , x , . . . , x n − λ − , x m − λ − , x m − λ , x n − λ + , . . . , x n − (cid:11) in the polynomial ring R = k [ x , x , . . . , x m ] . The closed subscheme in P m defined by the homogeneousideal K : = L ( λ ) ∩ J has Hilbert polynomial p + and corresponds to a point on the lexicographiccomponent of the Hilbert scheme Hilb p + ( P m ) .Proof. By Lemma 3.3, the Hilbert polynomial of the closed subscheme defined by the ideal L ( λ ) is p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . Proposition 3.5 implies that L ( λ ) + J = (cid:104) x , x , . . . , x m − (cid:105) . For allintegers j greater than 1, the sets { x m − λ − x j − m , x jn } and { x jm } form bases for the j -th homogeneouscomponents of R / J and R / (cid:0) L ( λ ) + J (cid:1) respectively. It follows that their Hilbert polynomials are theconstants 2 and 1. From the short exact sequence of graded R -modules0 RL ( λ ) ∩ J RL ( λ ) ⊕ RJ RL ( λ ) + J , we deduce that p + P m corresponding to themonomial ideal K = L ( λ ) ∩ J .Using Proposition 3.5, we also deduce that the saturation (cid:0) L ( λ ) : x ∞ m − (cid:1) is equal to the saturation ( K : x ∞ m − ) . Since both L ( λ ) and K are Borel-fixed ideals, Theorem 6 in [Ree95] establishes that thepoint corresponding to the homogeneous ideal K lies on the lexicographic component. (cid:3) Tangent spaces.
We show that these nearly lexicographic points are singular for a special classof integer partitions. The Zariski tangent space at the point in the Hilbert scheme Hilb p ( P m ) corresponding to the closed subscheme X in P m with ideal sheaf I X is naturally isomorphic toHom P m ( I X , O X ) = Hom X ( I X / I X , O X ) ; see [Kol96, Theorem 2.8]. Lemma 5.2.
Let λ : = (cid:0) ( m − ) r − s − , ( m − n ) s + (cid:1) be an integer partition where r − (cid:62) s (cid:62) andm − (cid:62) n (cid:62) . Set p ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) . The Hilbert scheme Hilb p + ( P m ) is singular at the pointcorresponding to the saturated monomial idealK : = x · (cid:10) x , x , . . . , x m − (cid:11) + x r − s − · (cid:10) x , x , . . . , x n − , x s + n (cid:11) = L ( λ ) ∩ (cid:10) x , x , x , . . . , x m − (cid:11) in the polynomial ring R = k [ x , x , . . . , x m ] .Proof. From the monomial generators for the lexicographic ideal L ( λ ) appearing in Proposition 3.5,we see that the given monomials generate the ideal K . It remains to show that the dimension of theZariski tangent space at the nearly lexicographic point is larger than the dimension of the Zariskitangent space at the lexicographic point. Equation (3.7.2) establishes that the dimension of later isless than or equal to (with equality holding when s > (cid:18) m + r − s − r − s − (cid:19) + (cid:18) m − n + s + s + (cid:19) + ( m − n + )( n − ) + m − . To estimate the dimension of the Zariski tangent space at the nearly lexicographic point, weexamine the sheaf on P m corresponding to the graded R -module Hom R ( K , R / K ) . Since the variable x m does not divide any of the generators of the ideal K , the dimension of this tangent space is MOOTH HILBERT SCHEMES 17 greater than or equal to dim k Hom R ( K , R / K ) ; see [RS97, Lemma 3.1]. Because K is a stablemonomial ideal, the Eliahou–Kervaire resolution [PS08, Theorem 2.3] yields a homogeneous freepresentation. The minimal syzygies among the generators of the ideal K are given by the blockmatrix Θ : = (cid:2) A A · · · A m − B B · · · B n − C (cid:3) where A T i : = x x x ··· x x i − x x i − x x i x x i + ··· x x m − x r − s x r − s − x ··· x r − s − x n − x r − s − x s + n · · · − x i x i − · · · · · · i · · · − x i + x i − · · · · · · i + ... ... . . . ... ... ... ... . . . ... ... ... . . . ... ... ... · · · − x m − · · · x i − · · · m − B T j : = x x x ··· x x m − x r − s ... x r − s − x j − x r − s − x j − x r − s − x j x r − s − x j + ... x r − s − x n − x r − s − x s + n . . . . . . − x j x j − · · · j . . . . . . − x j + x j − · · · j + ... ... . . . ... ... . . . ... ... ... ... . . . ... ... ... . . . · · · − x n − · · · x j − n − C T : = x x x ··· x x n − x x n x x n + ... x x m − x r − s x r − s − x ... x r − s − x n − x r − s − x s + n . . . − x r − x − x sn . . . . . . x . . . . . . − x s + n . . . x . . . . . . − x s + n . . . x ... ... . . . ... ... ... . . . ... ... ... . . . ... ... ... . . . . . . . . . − x s + n x n − m − for all 1 (cid:54) i (cid:54) m − (cid:54) j (cid:54) n −
1. It follows that Hom R ( K , R / K ) = Ker (cid:0)
Hom R ( Θ , R / K ) (cid:1) .The two ( m + n ) × x x x x x ··· x x m − x r − s − x r − s − x ··· x r − s − x n − x r − s − x s + n (cid:104) (cid:105) D T : = x x x · · · x n · · · (cid:104) (cid:105) D T : = · · · x x · · · x n − x s + n satisfy Θ T D = and Θ T D = . Thus, for all 2 (cid:54) i (cid:54) m , the column in the product x i D representsa nonzero element in Hom R ( K , R / K ) . The column in the product of the matrix D with anymonomial of degree r − s − x , x , . . . , x m (excluding x r − s − ) also represents anonzero element in Hom R ( K , R / K ) . There are ( m − ) + (cid:0) m + r − s − r − s − (cid:1) − x Θ lie in the ideal K , the m + n columns of the square matrix ··· m ... n − n x x m · · · · · · x x x m · · · · · · x x ... ... . . . ... ... ... . . . ... ... ... · · · x x m · · · x x m − · · · x x r − s − n · · · x r − s · · · x x r − s − n · · · x r − s − x ... ... . . . ... ... ... . . . ... ... ... · · · · · · x x r − s − n x r − s − x n − · · · · · · x x r − m x r − s − x s + n represent nonzero elements in Hom R ( K , R / K ) . Similarly, all of entries in the bottom n rows of thematrix Θ when multiplied by the monomial x r − s − lie in the ideal K . Hence, for all n (cid:54) j (cid:54) m , thecolumns of the matrices ... n − · · · x · · · x x ... ... . . . ... ... · · · x x m − x r − s − x j · · · x r − s x r − s − x j · · · x r − s − x ... ... . . . ... ... · · · x r − s − x j x r − s − x n − · · · x r − s − x s + n , ... (cid:0) m − n + s + s + (cid:1) − · · · x · · · x x ... ... . . . ... ... · · · x x m − · · · x r − s · · · x r − s − x ... ... . . . ... ... · · · x r − s − x n − x r − s − x sn x n + x r − s − x sn x n + · · · x r − s − x s + m x r − s − x s + n represent nonzero elements in Hom R ( K , R / K ) . Each nonzero entry in the bottom row of the secondmatrix is the product of x r − s − and a monomial of degree s + x n , x n + , . . . , x m (excluding x s + n ). Hence, there are (cid:0) m − n + s + s + (cid:1) − R ( K , R / K ) is N : = ( m − ) + (cid:18) m + r − s − r − s − (cid:19) − + ( m + n ) + ( m − n + )( n − ) + (cid:18) m − n + s + s + (cid:19) − . By comparing their nonzero monomial entries, we see that these N columns are linearly independent.The difference between the number N and (5.2.4) is n −
1. As n (cid:62)
2, we conclude that the Hilbertscheme is singular at the point corresponding to the monomial ideal K . (cid:3) Proposition 5.3.
Let λ : = ( λ , λ , . . . , λ r ) be an integer partition such that λ ∪ ( ) has at least threedistinct parts. Fix m > λ , set p ( t ) : = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) , and consider the monomial idealJ : = (cid:10) x , x , . . . , x m − λ − , x m − λ − , x m − λ , x m − λ + , . . . , x m − (cid:11) . in the polynomial ring R = k [ x , x , . . . , x m ] . The Hilbert scheme Hilb p + ( P m ) is singular at thepoint corresponding to the saturated monomial ideal K : = L ( λ ) ∩ J.Proof.
Lemma 5.1 shows that the nearly lexicographic point lies on the lexicographic component ofthe Hilbert scheme Hilb p + ( P m ) . We reduce the analysis to a special case.The inclusion (cid:104) x , x , . . . , x m − λ − (cid:105) ⊂ K implies that the nearly lexicographic point is containedin a ( λ + ) -plane. Hence, we may assume that m = λ + L (cid:0) λ ∪ ( ) (cid:1) is the flat limit of a one-parameter family whose generalmember is the sum of the lexicographic ideal L ( λ ) and the ideal of a disjoint point. Since thedimension of the Zariski tangent space at a point in family is an upper-semicontinuous function, wemay also assume that λ r > L ( λ ∪ ( )) determines a residual flag ∅ ⊂ X e ⊂ X e − ⊂ · · · ⊂ X in P m . The hypotheses ensure that e (cid:62)
3. The closed subscheme X e − lies in some linear space Λ ⊆ P m . We may deform the scheme X e − in the linear space Λ and leave the rest of the residual MOOTH HILBERT SCHEMES 19 flag X e − ⊂ X e − ⊂ · · · ⊂ X unchanged. If the closed scheme X e − corresponds to a singular pointon the Hilbert scheme in Λ , then it follows that the closed scheme X corresponds to a singular pointon the Hilbert scheme. Thus, we may assume that e = λ = (cid:0) ( m − ) r − s − , ( m − n ) s + (cid:1) where r − (cid:62) s (cid:62) m − (cid:62) n (cid:62)
2. In this special case, Lemma 5.2 proves that the dimension ofthe Zariski tangent space at the nearly lexicographic point exceeds the dimension of the lexicographiccomponent. Therefore, the Hilbert scheme Hilb p + ( P m ) is singular at the point corresponding tothe saturated monomial ideal K : = L ( λ ) ∩ J . (cid:3) Other singular examples.
In addition to our family of singular Hilbert schemes, the classificationof smooth Hilbert schemes relies on three other singular families.
Example 5.4.
Two familiar Hilbert schemes explain the curious gap between conditions (4) and(5) in Proposition 4.9. By appealing to the splitting in (4.8.3), it is enough to understand integerpartitions ( , ) , ( , ) and ( , ) . The first of these is already covered by both Theorem 4.2 andExample 4.11. In contrast, the Hilbert schemes in the other two cases are singular.The integer partition ( , ) is associated to the Hilbert polynomial 2 t +
2. The Hilbert schemeHilb t + ( P m ) is singular; it has two irreducible components. A general point on one componentcorresponds to a pair of skew lines and a general point on the other corresponds to the union of aplane conic and an isolated point; compare with [CCN11, Theorem 1.1].The integer partition ( , ) is associated to the Hilbert polynomial 3 t +
1. The Hilbert schemeHilb t + ( P m ) is again singular because it has two irreducible components. A general point in firstcomponent corresponds to a twisted cubic curve and a general point in the other corresponds to theunion of a plane cubic and an isolated points; compare with [PS85, Theorem]. (cid:5) Example 5.5.
For completeness, we also add an explicit description of another well-known singularHilbert scheme. For any nonnegative integer s , consider the integer partition λ = ( s + ) whoseassociated Hilbert polynomial is the constant s +
4. For all m (cid:62)
3, the Hilbert scheme Hilb s + ( P m ) is singular at the point corresponding to the saturated homogeneous ideal B ( s ) : = (cid:10) x , x , . . . , x m − , x m − , x m − x m − , x m − x m − , x m − , x m − x m − , x s + m − (cid:11) in the polynomial ring R = k [ x , x , . . . , x m ] ; compare with [Che98, Lemma 1.4]. (cid:5) Theorem 5.6.
Let E be a locally free sheaf on a locally noetherian scheme S of constant rank m + and let p be a numerical polynomial. The Hilbert scheme Hilb p ( P ( E )) is smooth over S if and onlyif there exists an integer partition λ = ( λ , λ , . . . , λ r ) such that p ( t ) = ∑ ri = (cid:0) t + λ i − i λ i − (cid:1) and one of thefollowing seven conditions holds: (1) m = , (2) λ r (cid:62) , (3) λ = ( ) or λ = ( m r − , λ r − , ) = ( r − (cid:122) (cid:125)(cid:124) (cid:123) m , m , . . . , m , λ r − , ) where r (cid:62) and m (cid:62) λ r − (cid:62) , (4) λ = ( m r − s − , λ s + r − s − , ) where r − (cid:62) s (cid:62) and m − (cid:62) λ r − s − (cid:62) , (5) λ = ( m r − s − , s + , ) where r − (cid:62) s (cid:62) , (6) λ = ( m r − , ) where r (cid:62) , (7) λ = ( m + ) or r = . Proof.
Remark 4.12 already shows that each condition implies that the Hilbert scheme is smooth.Hence, it suffices to prove that the Hilbert scheme has a singular point when the integer partition λ : = ( λ , λ , . . . , λ r ) does not satisfy conditions (1) – (7). To bypass conditions (1) and (2), we musthave m (cid:62) λ r =
1. By Lemma 4.1, we may assume that S is the spectrum of an algebraicallyclosed field. Using the splitting in (4.8.3), we may also assume that m > λ . For the remaininginteger partitions with one distinct part, Example 5.5 describes a singularity. When the integerpartition has two distinct parts, there are two outstanding cases, namely λ = ( , ) or ( , ) , andExample 5.4 exhibits their singularities. Finally, Proposition 5.3 identifies a singular point wheneverthe integer partition has at least three distinct parts. (cid:3) Acknowledgements.
We thank Dave Anderson, Sam Payne, Mike Roth, and Michael E. Stillmanfor their suggestions. Computational experiments done in
Macaulay2 [M2] were indispensable.GGS was partially supported by the Natural Sciences and Engineering Research Council of Canada(NSERC) and the Knut and Alice Wallenberg Foundation.R
EFERENCES [Bea83] Arnaud Beauville,
Variétés Kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. (1983), no. 4, 755–782.[BH93] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings , Cambridge Studies in Advanced Mathematics,vol. 39, Cambridge University Press, Cambridge, 1993.[CCN11] Dawei Chen, Izzet Coskun, and Scott Nollet,
Hilbert scheme of a pair of codimension two linear subspaces ,Comm. Algebra (2011), no. 8, 3021–3043.[Che98] Jan Cheah, Cellular decompositions for nested Hilbert schemes of points , Pacific J. Math. (1998), no. 1,39–90.[EGA71] Alexander Grothendieck and Jean A. Dieudonné,
Eléments de géométrie algébrique I , 2nd ed., Grundlehrender Mathematischen Wissenschaften, vol. 166, Springer-Verlag, Berlin, 1971.[EHM92] Philippe Ellia, André Hirschowitz, and Emilia Mezzetti,
On the number of irreducible components of theHilbert scheme of smooth space curves , Internat. J. Math. (1992), no. 6, 799–807.[Fog68] John Fogarty, Algebraic families on an algebraic surface , Amer. J. Math (1968), 511–521.[Ful98] William Fulton, Intersection theory , 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd Series,vol. 2, Springer-Verlag, Berlin, 1998.[GKP94] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik,
Concrete mathematics , 2nd ed., Addison-WesleyPublishing Company, Reading, MA, 1994.[Got78] Gerd Gotzmann,
Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes , Math.Z. (1978), no. 1, 61–70.[Gro96] I. Grojnowski,
Instantons and affine algebras. I. The Hilbert scheme and vertex operators , Math. Res. Lett. (1996), no. 2, 275–291.[Hai01] Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture , J. Amer. Math. Soc. (2001), no. 4, 941–1006.[Har66] Robin Hartshorne, Connectedness of the Hilbert scheme , Inst. Hautes Études Sci. Publ. Math. (1966),5–48.[KM85] Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves , Annals of Mathematics Studies,vol. 108, Princeton University Press, Princeton, NJ, 1985.[Kol96] János Kollár,
Rational curves on algebraic varieties , Ergebnisse der Mathematik und ihrer Grenzgebiete.3rd Series., vol. 32, Springer-Verlag, Berlin, 1996.[Lin09] Kevin H. Lin,
When are Hilbert schemes smooth? , MathOverflow, Question 244, 2009.
MOOTH HILBERT SCHEMES 21 [M2] Daniel R. Grayson and Michael E. Stillman,
Macaulay2, a software system for research in algebraicgeometry , available at .[Mac15] Ian G. Macdonald,
Symmetric functions and Hall polynomials , 2nd ed., Oxford Classic Texts in the PhysicalSciences, The Clarendon Press, Oxford University Press, New York, 2015.[Mac27] Francis S. Macaulay,
Some Properties of Enumeration in the Theory of Modular Systems , Proc. LondonMath. Soc. (2) (1927), 531–555.[Mum62] David Mumford, Further pathologies in algebraic geometry , Amer. J. Math. (1962), 642–648.[Nak97] Hiraku Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces , Ann. of Math.(2) (1997), no. 2, 379–388.[PS08] Irena Peeva and Michael E. Stillman,
The minimal free resolution of a Borel ideal , Expo. Math. (2008),no. 3, 237–247.[PS85] Ragni Piene and Michael Schlessinger, On the Hilbert scheme compactification of the space of twistedcubics , Amer. J. Math. (1985), no. 4, 761–774.[Ram19] Ritvik Ramkumar,
Hilbert schemes with few Borel fixed points (2019), available at arXiv:1907.13335 .[Ree95] Alyson A. Reeves,
The radius of the Hilbert scheme , J. Algebraic Geom. (1995), no. 4, 639–657.[RS97] Alyson A. Reeves and Michael E. Stillman, Smoothness of the lexicographic point , J. Algebraic Geom. (1997), no. 2, 235–246.[Sta20] Andrew Staal, The ubiquity of smooth Hilbert schemes , Math. Z. (2020), to appear.[Sta19] ,
Hilbert schemes with two Borel-fixed points in arbitrary characteristic (2019), preprint.[Vak06] Ravi Vakil,
Murphy’s law in algebraic geometry: badly-behaved deformation spaces , Invent. Math. (2006), no. 3, 569–590.R OY S KJELNES : D
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