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To make a quantitative statement about all Hilbert schemes, we interpret Macaulay’s classification as follows: First, identify any admissible Hilbert polynomial p with its sequence b = (b1, b2,...,br). These sequences are generated by two operations, namely “integrating” p to Φ(p) := b + (1, 1,..., 1) and “adding one” to p to get A(p) := 1 + p = (b, 0). The set of all such sequences forms an infinite full binary tree. There is an associated tree, which p n we denote Hc, whose vertices are the Hilbert schemes Hilb (P ) parametrizing codimension c = n − deg p subschemes, for each positive c ∈ Z. Geometrically, Φ corresponds to coning over parametrized schemes and A to adding a point. We endow Hc with a natural probability distribution, in which the vertices at a fixed height are equally likely. This leads to our second main result. Theorem 1.2. Let K be algebraically closed or have characteristic 0. The probability that a random Hilbert scheme is irreducible and nonsingular is greater than 0.5. This theorem counterintuitively suggests that the geometry of the majority of Hilbert schemes is understandable. To prove Theorems 1.1 and 1.2, we study the algorithm generating saturated strongly stable ideals first described by Reeves [Ree92] and later generalized in [Moo12, CLMR11]. We obtain precise information about Hilbert series and K-polynomials of saturated strongly stable ideals. The primary technical result we need is the following.
Theorem 1.3. Let I ⊂ K[x0, x1,...,xn] be a saturated strongly stable ideal with Hilbert p polynomial p, let Ln be the corresponding lexicographic ideal in K[x0, x1,...,xn], and let KI p p be the numerator of the Hilbert series of I. If I =6 Ln, then we have deg KI < deg KLn . The structure of the paper is as follows. In Section 2.1, we introduce two binary relations on the set of admissible Hilbert polynomials and show that they generate all such polynomials. The set of lexicographic ideals is then partitioned by codimension into infinitely many binary trees in Section 2.2. Geometrically, these are trees of Hilbert schemes, as every Hilbert scheme contains a unique lexicographic ideal. To identify well-behaved Hilbert schemes, we review saturated strongly stable ideals in Section 3 and we examine their K-polynomials in Section 4. The main results are in Section 5.
Conventions. Throughout, K is a field, N is the nonnegative integers, and K[x0, x1,...,xn] is the standard Z-graded polynomial ring. The Hilbert function, polynomial, series, and K-polynomial of the quotient K[x0, x1,...,xn]/I by a homogeneous ideal I are denoted hI , pI, HI , and KI , respectively.
Acknowledgments. We especially thank Gregory G. Smith for his guidance in this re- search. We thank Mike Roth, Ivan Dimitrov, Tony Geramita, Chris Dionne, Ilia Smirnov, Nathan Grieve, Andrew Fiori, Simon Rose, and Alex Duncan for many discussions. We also thank the anonymous referee for helpful remarks that improved the paper. This research was supported by an E.G. Bauman Fellowship in 2011-12, by Ontario Graduate Scholarships in 2012-15, and by Gregory G. Smith’s NSERC Discovery Grant in 2015-16. 2 2. Binary Trees and Hilbert Schemes In 2.1 we observe that a tree structure exists on the set of numerical polynomials de- termining nonempty Hilbert schemes. Macaulay’s pioneering work [Mac27] classifies these polynomials and two mappings turn this set into an infinite binary tree. In 2.2 we find related binary trees in the sets of lexicographic ideals and Hilbert schemes.
2.1. The Macaulay Tree. Let K be a field and let K[x0, x1,...,xn] denote the homoge- n neous (standard Z-graded) coordinate ring of n-dimensional projective space PK. Let M be a finitely generated graded K[x0, x1,...,xn]-module. The Hilbert function hM : Z → Z of M is defined by hM (i) := dimK(Mi) for all i ∈ Z. Every such M has a Hilbert polynomial pM , that is, a polynomial pM (t) ∈ Q[t] such that hM (i)= pM (i) for i ≫ 0; see [BH93, Theo- rem 4.1.3]. For a homogeneous ideal I ⊂ K[x0, x1,...,xn], let hI and pI denote the Hilbert function and Hilbert polynomial of the quotient module K[x0, x1,...,xn]/I, respectively. We begin with a basic example and make a notational convention, for later use. Example 2.1. Fix a nonnegative integer n. By the classic stars-and-bars argument [Sta12, i+n Section 1.2], we have hS(i) = n for all i ∈ Z, where S = K[x0, x1,...,xn]. The equality h (i)= p (i) is only valid for i ≥ −n, because the polynomial p (t)= t+n ∈ Q[t] only has S S S n roots −n, −n +1,..., −1, whereas h (i) = 0 for all i< 0. S j j! j Remark 2.2. For integers j, k we set k = k!(j−k)! if j ≥ k ≥ 0 and k = 0 otherwise. For a Z t+a (t+a)(t+a−1)···(t+a−b+1) Q t+a variable t and a, b ∈ , we define b = b! ∈ [t ] if b ≥ 0, and b =0 otherwise. When b ≥ 0, the polynomial t+a has degree b with zeros −a, −a+1,..., −a+b−1. b Importantly, we have t+a | =6 j+a = 0 when j < −a. Interestingly, [Mac27, p.533] uses b t=j b distinct notation for polynomial and integer binomial coefficients. A polynomial is an admissible Hilbert polynomial if it is the Hilbert polynomial of some closed subscheme in some Pn. Admissible Hilbert polynomials correspond to nonempty Hilbert schemes. We use the well-known classification first discovered by Macaulay. Proposition 2.3. The following conditions are equivalent: (i) The polynomial p(t) ∈ Q[t] is a nonzero admissible Hilbert polynomial. d t+i t+i−ei (ii) There exist integers e0 ≥ e1 ≥···≥ ed > 0 such that p(t)= i=0 i+1 − i+1 . r t+bj −j+1 (iii) There exist integers b1 ≥ b2 ≥···≥ br ≥ 0 such that p(t)= . Pj=1 bj Moreover, these correspondences are bijective. P Proof. (i) ⇔ (ii) This is proved in [Mac27]; see the formula for “χ(ℓ)” at the bottom of p.536. For a geometric account, see [Har66, Corollary 3.3 and Corollary 5.7]. (i) ⇔ (iii) This follows from [Got78, Erinnerung 2.4]; see also [BH93, Exercise 4.2.17]. Uniqueness of the sequences of integers also follows. For simplicity, we always work with nonzero admissible Hilbert polynomials. Let the Macaulay–Hartshorne expression of an admissible Hilbert polynomial p be its expres- d t+i t+i−ei sion p(t)= i=0 i+1 − i+1 , for e0 ≥ e1 ≥···≥ ed > 0, and the Gotzmann expression r t+bj −j+1 of p be its expression p(t)= , for b1 ≥ b2 ≥···≥ br ≥ 0. From these, we find P j=1 bj the degree d = b1, the leading coefficient ed/d!, and the Gotzmann number r of p, which P 3 bounds the Castelnuovo–Mumford regularity of saturated ideals with Hilbert polynomial p. In particular, such ideals are generated in degree r; see [IK99, p. 300-301]. Macaulay–Hartshorne and Gotzmann expressions are conjugate. Recall that the conju- k gate partition to a partition λ =(λ1,λ2,...,λk) of an integer ℓ = i=1 λi is the partition of ℓ obtained from the Ferrers diagram of λ by interchanging rows and columns, having λi −λi+1 parts equal to i; see [Sta12, Section 1.8]. P Lemma 2.4. If p(t) ∈ Q[t] is an admissible Hilbert polynomial with Macaulay–Hartshorne d t+i t+i−ei expression i=0 i+1 − i+1 for e0 ≥ e1 ≥ ··· ≥ ed > 0 and Gotzmann expression r t+bj −j+1 for b1 ≥ b2 ≥ ··· ≥ br ≥ 0, then r = e0 and the nonnegative partition j=1 bjP (b , b ,...,b ) is conjugate to the partition (e , e ,...,e ). P1 2 r 1 2 d d t+i t+i−ed d−1 t+i−ed t+i−ei Proof. The key step is to rewrite p as i=0 i+1 − i+1 + i=0 i+1 − i+1 and to prove d t+i − t+i−ed = ed t+d−j+1 , by induction on d. This gives the expression i=0 i+1 i+1 j=1 Pd P e P d t +d −Pj +1 d−1 s + i s + i − e + e p(t)= + − i d d i +1 i +1 j=1 " i=0 # X X s=t−ed and one can iterate on the second part. So r = e0 and the partition (b1, b2,...,br) has ei −ei+1 r d d parts equal to i, for all 0 ≤ i ≤ d. The equalities j=1 bj = i=0(ei − ei+1)i = i=1 ei then show that (b , b ,...,b ) is conjugate to (e , e ,...,e ). 1 2 r 1 2 P d P P For p(t) as in Lemma 2.4, it is convenient to refer to (e0, e1,...,ed) as its Macaulay– Hartshorne partition and (b1, b2,...,br) as its (nonnegative) Gotzmann partition. We now describe two fundamental binary relations on admissible Hilbert polynomials. The first takes the polynomial p with partitions e =(e0, e1,...,ed) and b =(b1, b2,...,br) to the polynomial Φ(p) with partitions (e0, e) and b + (1, 1,..., 1) (add one to each entry). The second takes p to A(p)=1+ p, with partitions e + (1, 0,..., 0) and (b, 0). Both Φ(p) and A(p) are admissible by Proposition 2.3. The backwards difference operator ∇ maps any q ∈ Q[t] to q(t)−q(t−1). Backwards differences are discrete derivatives—in Lemma 2.5, (ii) says that Φ is the indefinite integral and (iii) is a well-known discrete analogue of the Fundamental Theorem of Calculus. Lemma 2.5. If p(t) is an admissible Hilbert polynomial with Macaulay–Hartshorne partition (e0, e1,...,ed) and Gotzmann partition (b1, b2,...,br), then the following hold: (i) [∇(p)](t)= r t+bj −1−j+1 = d−1 t+i − t+i−ei+1 ; j=1 bj −1 i=0 i+1 i+1 (ii) ∇AaΦ(p)= p, for all a ∈ N; and P P (iii) if deg p > 0 and k ∈ {1, 2,...,r} is the largest index such that bk =6 0, then we have p − Φ∇(p)= r − k, but if deg p =0, then ∇(p)=0. Proof. These follow by linearity of ∇ and the binomial addition formula. We now observe that the set of admissible Hilbert polynomials forms a tree. Proposition 2.6. The tree with vertices corresponding to admissible Hilbert polynomials and edges corresponding to pairs of the form p, A(p) and p, Φ(p) , for all admissible Hilbert polynomials p, forms an infinite full binary tree. The root of the tree corresponds to 1. We call this the Macaulay tree M. It has 2j vertices at height j, for all j ∈ N. 4 r Proof. By induction on r, p(t) = t+bj −j+1 = Φbr AΦbr−1−br A ··· AΦb2−b3 AΦb1−b2 (1) j=1 bj holds. P A portion of M is displayed in Figure 1, in terms of Gotzmann expressions.
t t−1 0 + 0 + t−2 + 0 t−3 + t−4 0 0 t + t−1 + 0 0 t−2 + t−3 0 0 t+1 t 1 + 1 + t−1 + t−2 1 1 t t−1 t−2 0 + 0 + 0
t+1 t 1 + 1 + t−1 + t−3 1 0 t+1 t t−1 1 + 1 + 1 t+2 t+1 t 2 + 2 + 2
t t−1 0 + 0 t+1 t 1 + 1 + t−2 + t−3 0 0 t+1 t t−2 1 + 1 + 0 t+2 2 + t+1 + t−1 A 2 1 t+1 t 1 + 1 t+2 2 + t+1 + t−2 2 0 t+2 t+1 2 + 2 t+3 t+2 t 3 + 3 0 t+1 t−1 1 + 0 + t−2 + t−3 t+1 0 0 1 + t−1 + t−2 0 0 t+2 t t−1 2 + 1 + 1 Φ t+1 t−1 1 + 0 t+2 t t−2 2 + 1 + 0 t+2 t 2 + 1 t+3 t+1 3 + 2
t+1 1 t+2 + 2 t−1 + t−2 0 0 t+2 t−1 2 + 0 t+3 t 3 + 1 t+2 2 t+3 t−1 3 + 0 t+3 3 t+4 4 Figure 1. The Macaulay tree M to height 4 with Gotzmann expressions
M d t+i t+i−ei Remark 2.7. The path from the root 1 of the tree to p(t) = i=0 i+1 − i+1 can also be expressed as p = Ae0−e1 ΦAe1−e2 Φ ··· ΦAed−1−ed ΦAed−1(1). 5 P Example 2.8. The Hilbert polynomial 3t+1 of the twisted cubic curve X ⊂ P3 has partitions (b1, b2, b3, b4)=(1, 1, 1, 0) and (e0, e1)=(4, 3). The path in M from 1 to 3t+1 can be written as Φ0AΦ1−0AΦ1−1AΦ1−1(1) = A4−3ΦA3−1(1) = AΦA2(1). This path is shown in Figure 2.
5 4 4t − 2 3 3t + 1 Φ A A 3t (3/2)t2+(3/2)t+1 2 2t + 3 2t + 2 t2 + 3t A 2t + 1 t2 +2t +2
t2 +2t +1 (1/3)t3+(3/2)t2+ (13/6)t +1
1 t + 4 t + 3 (1/2)t2 + (7/2)t t + 2 (1/2)t2+(5/2)t+2
(1/2)t2+(5/2)t+1 (1/6)t3+(3/2)t2+ (7/3)t + 1
t + 1 (1/2)t2+(3/2)t+3
(1/2)t2+(3/2)t+2 (1/6)t3 + t2 + (17/6)t +1
(1/2)t2+(3/2)t+1 (1/6)t3 + t2 + (11/6)t +2
(1/6)t3 + t2 + (11/6)t +1 (1/24)t4 + (5/12)t3 + (35/24)t2 + (25/12)t +1
Figure 2. The path from 1 to p(t)=3t + 1 in the Macaulay tree
2.2. Lexicographic and Hilbert Trees. We now connect lexicographic ideals and Hilbert schemes to the Macaulay tree M. Specifically, M reappears infinitely many times in the set of saturated lexicographic ideals and the set of Hilbert schemes, with one tree for each positive codimension. Two mappings on lexicographic ideals analogous to Φ and A are essential. n+1 u u0 u1 un For any vector u = (u0,u1,...,un) ∈ N , let x = x0 x1 ··· xn . The lexicographic u v ordering is the relation ≻ on the monomials in K[x0, x1,...,xn] defined by x ≻ x if the first nonzero coordinate of u − v ∈ Zn+1 is positive, where u, v ∈ Nn+1. 6 Example 2.9. We have x0 ≻ x1 ≻···≻ xn in lexicographic order on K[x0, x1,...,xn]. If 2 4 3 n ≥ 2, then x0x2 ≻ x1 ≻ x1. Lexicographic ideals are monomial ideals whose homogeneous pieces are spanned by max- imal monomials in lexicographic order. For a homogeneous ideal I in K[x0, x1,...,xn], lexicographic order gives rise to two monomial ideals associated to I. First, the lexico- hI graphic ideal for the Hilbert function hI in K[x0, x1,...,xn] is the monomial ideal Ln whose ith graded piece is spanned by the dimK Ii = hK[x0,x1,...,xn](i) − hI (i) largest monomials hI in K[x0, x1,...,xn]i, for all i ∈ Z. The equality hI = h hI holds by definition and L is a Ln n homogeneous ideal of K[x0, x1,...,xn]; see [Mac27, § II] or [MS05b, Proposition 2.21]. More importantly, the (saturated) lexicographic ideal for the Hilbert polynomial pI is the monomial ideal
pI hI ∞ j hI Ln := Ln : hx0, x1,...,xni = f ∈ K[x0, x1,...,xn] | fhx0, x1,...,xni ⊆ Ln . j≥1 [ Saturation with respect to the irrelevant ideal hx0, x1,...,xni ⊂ K[x0, x1,...,xn] does not pI affect the Hilbert function in large degrees, so Ln also has Hilbert polynomial pI . From here on, we essentially always work with saturated lexicographic ideals and point out when this is not the case. Given a finite sequence of nonnegative integers a0, a1,...,an−1 ∈ N, consider the monomial ideal L(a0, a1,...,an−1) ⊂ K[x0, x1,...,xn] from [RS97, Notation 1.2] with generators
an−1+1 an−1 an−2+1 an−1 an−2 a2 a1+1 an−1 an−2 a1 a0 hx0 , x0 x1 ,...,x0 x1 ··· xn−3xn−2 , x0 x1 ··· xn−2xn−1i. Lemma 2.10(i) appears in [Moo12, Theorem 2.23].
d t+i t+i−ei Lemma 2.10. Let p(t)= i=0 i+1 − i+1 , for integers e0 ≥ e1 ≥···≥ ed > 0, and let n ∈ N satisfy n>d = deg p. P (i) Define ei =0, for d +1 ≤ i ≤ n, and aj = ej − ej+1, for all 0 ≤ j ≤ n − 1. We have p Ln = L(a0, a1,...,an−1)
ad+1 = hx0, x1,...,xn−d−2, xn−d−1,
ad ad−1+1 ad ad−1 a2 a1+1 ad ad−1 a1 a0 xn−d−1xn−d ,...,xn−d−1xn−d ··· xn−3xn−2 , xn−d−1xn−d ··· xn−2xn−1i.
(ii) If there is an integer 0 ≤ ℓ ≤ d − 1 such that aj =0 for all j ≤ ℓ, and aℓ+1 > 0, then p the minimal monomial generators of Ln are given by m1, m2,...,mn−ℓ−1, where
mi = xi−1, for all 1 ≤ i ≤ n − d − 1, k−1 ad−j ad−k+1 mn−d+k = xn−d−1+j xn−d−1+k , for all 0 ≤ k ≤ d − ℓ − 2, and j=0 ! Y d−ℓ−1 ad−j mn−ℓ−1 = xn−d−1+j. j=0 Y If a0 =06 , then the minimal monomial generators are those listed in (i). Proof.
(i) It is straightforward to check that the ideal L = L(a0, a1,...,an−1) is saturated and lexicographic. To see that L has the correct Hilbert polynomial, one can first prove 7 ′ d t+i t+i−ad that L = L(0,..., 0, ad, 0,..., 0) has Hilbert polynomial i=0 i+1 − i+1 . The general case then follows by induction on d = deg p and the short exact sequence ′′ P ′ 0 → (K[x0, x1,...,xn]/L )(−ad) → K[x0, x1,...,xn]/L → K[x0, x1,...,xn]/L → 0, ′′ ad where L = L(a0, a1,...,ad−1, 0, 0,..., 0) and the injection sends 1 7→ xn−d−1. ad ad−1 a1 a0 ad ad−1 aℓ+2 aℓ+1 (ii) We know that xn−d−1xn−d ··· xn−2xn−1 = xn−d−1xn−d ··· xn−ℓ−3xn−ℓ−2, because either a0 = a1 = ··· = aℓ = 0, or a0 =6 0 and ℓ = −1. If ℓ ≥ 0, then the monomial generators
ad ad−1 aℓ+2 aℓ+1+1 ad ad−1 aℓ+1 aℓ+1 xn−d−1xn−d ··· xn−ℓ−3xn−ℓ−2 , xn−d−1xn−d ··· xn−ℓ−2xn−ℓ−1,..., ad ad−1 a2 a1+1 xn−d−1xn−d ··· xn−3xn−2 from (i) are redundant, as they are multiples of the last monomial generator. Removing these gives the monomial generators m1, m2,...,mn−ℓ−1. For all 1
In analogy with Φ, for any ideal I ⊆ K[x0, x1,...,xn], we denote the extension ideal by Φ(I)= I · K[x0, x1,...,xn+1]. The following is similar to Lemma 2.12. Proposition 2.13. Let p be an admissible Hilbert polynomial and n> deg p an integer. We p Φ(p) have Φ Ln = Ln+1 . Equivalently Φ L(a0, a1,...,an−1) = L(0, a0, a1,...,an−1) holds, for all a , a ,...,a ∈ N, and extension preserves codimension. 0 1 n−1 ′ ′ Proof. This follows by relabelling a0 = 0 and ai = ai−1, for 0 < i ≤ d + 1, on generators. p n p n The lexicographic point of Hilb (P ) is the point XLn , where XI ⊆ P denotes the closed subscheme with saturated ideal I. The lexicographic point is nonsingular and lies on a unique irreducible component Hilbp(Pn) called the lexicographic component [RS97]. We can now describe a tree structure on the set of Hilbert schemes. We regard this as a rough chart of the geography of Hilbert schemes, developed further in Section 5.
Theorem 2.14. For each positive codimension c ∈ Z, the graph Hc whose vertex set con- sists of all nonempty Hilbert schemes Hilbp(Pn) parametrizing codimension c subschemes and whose edges are all pairs Hilbp(Pn), HilbA(p)(Pn) and Hilbp(Pn), HilbΦ(p)(Pn+1) , where 8 p is an admissible Hilbert polynomial and n = c + deg p, is an infinite full binary tree. The 1 c c root of the tree Hc is the Hilbert scheme Hilb (P )= P . Proof. Each pair of an admissible Hilbert polynomial p and positive c ∈ Z uniquely deter- p p c+deg p mines L = Lc+deg p and the Hilbert scheme Hilb (P ) containing [XL]. Lemma 2.12 and Proposition 2.13 combined with Lemma 2.10 show that the mapping p 7→ Hilbp(Pc+deg(p)) is a graph isomorphism. The root is then Hilb1(Pc)= Pc. For positive c ∈ Z, we call the tree of Theorem 2.14 the Hilbert tree of codimension H H H c, denoted c. We call the disjoint union = c∈N,c>0 c the Hilbert forest.
3. Strongly StableF Ideals This section reviews a well-known algorithm that generates saturated strongly stable ideals and then examines analogues of Φ and A for strongly stable ideals. A monomial ideal I ⊆ K[x0, x1,...,xn] is strongly stable,or0-Borel, if, for all monomi- −1 als m ∈ I, for all xj dividing m, and for all xi ≻ xj, we have xj mxi ∈ I. In characteristic 0, n this is equivalent to being Borel-fixed, i.e. fixed by the action γ · xj = i=0 γijxi of upper triangular matrices γ ∈ GLn+1(K)[BS87a, Proposition 2.7]. For any monomial m, let max m P be the maximum index j such that xj divides m, and min m be the minimum such index. 2 2 Example 3.1. The monomial ideal I = hx0, x0x1, x1i ⊂ K[x0, x1, x2] is strongly stable. The 5 2 monomial m = x1x2x7 ∈ K[x0, x1,...,x13] satisfies max m = 7 and min m = 1. For a monomial ideal I, let G(I) denote its minimal set of monomial generators. We gather some useful properties of strongly stable ideals in the following lemma.
Lemma 3.2. Let I ⊆ K[x0, x1,...,xn] be a monomial ideal.
(i) The ideal I is strongly stable if and only if, for all g ∈ G(I), for all xj dividing g, and −1 for all xi ≻ xj, we have gxixj ∈ I. ′ ′ ′ −1 (ii) If, for all g ∈ G(I), for all xj dividing g , and for all xi ≻ xj, we have g xixj ∈ I, then, for all monomials m ∈ I, there exists a unique g ∈ G(I) and unique monomial ′ ′ ′ m ∈ K[x0, x1,...,xn] such that m = gm and max g ≤ min m . (iii) If I is a strongly stable ideal, then I is saturated with respect to the irrelevant ideal hx0, x1,...,xni if and only if no minimal monomial generators of I are divisible by xn. (iv) If I is strongly stable with constant Hilbert polynomial pI ∈ N, then there exists an k integer k ∈ N such that xn−1 ∈ I. Proof. (i) If I is strongly stable, then this holds for all g ∈ I. Conversely, let m ∈ I be any monomial. By (ii), there is a unique factorization m = gm′, where g ∈ G(I) and ′ ′ m ∈ K[x0, x1,...,xn] is a monomial such that max g ≤ min m . Let xj divide m and −1 −1 let xi ≻ xj. Either xj divides g, in which case gxixj ∈ I and mxixj ∈ I, or xj divides ′ −1 m , in which case mxixj is a multiple of g. (ii) See the proof of [MS05b, Lemma 2.11]. −1 (iii) If xn divides a minimal generator g ∈ G(I), then for all xj we have (gxn )xj ∈ I, while −1 gxn ∈/ I. Conversely, any monomial m ∈ (I : hx0, x1,...,xni) \ I yields a minimal monomial generator mxn ∈ G(I), by (ii). (iv) See the proof of [Moo12, Lemma 3.17]. 9 Strongly stable ideals are generated by expansions. Let I ⊆ K[x0, x1,...,xn] be a saturated strongly stable ideal. A generator g ∈ G(I) is expandable if there are no minimal monomial −1 generators of I in the set xi gxi+1 | xi divides g and 0 ≤ i < n − 1 . The expansion of I at an expandable generator g is the monomial ideal ′ I = hI \hgii + hgxj | max g ≤ j ≤ n − 1i ⊂ K[x0, x1,...,xn]; see [Moo12, Definition 3.4]. The monomial 1 ∈ h1i is vacuously expandable with expan- sion hx0, x1,...,xn−1i ⊂ K[x0, x1,...,xn]. Parts (i) and (iii) of Lemma 3.2 ensure that the expansion of a saturated strongly stable ideal is again saturated and strongly stable. p Example 3.3. Let Ln = L(a0, a1,...,an−1) = hm1, m2,...,mn−ℓ−1i be lexicographic, as in ad ad−1 aℓ+2 aℓ+1 Lemma 2.10. The last generator is mn−ℓ−1 = xn−d−1xn−d ··· xn−ℓ−3xn−ℓ−2. If ai > 0, then −1 xn−i−1 divides mn−ℓ−1xn−ixn−i−1 to order ai − 1, which is not the case for any mj. Therefore, the expansion at mn−ℓ−1 has generators
{m1, m2,...,mn−ℓ−2, mn−ℓ−1xn−ℓ−2, mn−ℓ−1xn−ℓ−1,...,mn−ℓ−1xn−1}. p These are easily verified to be minimal generators of L(a0 +1, a1, a2,...,an−1)= A Ln .
For a saturated strongly stable ideal I ⊆ K[x0, x1,...,xn], let ∇ (I) ⊆ K[x0, x1,...,x n−1] be the image of I under the mapping K[x0, x1,...,xn] → K[x0, x1,...,xn−1], defined by xj 7→ xj for 0 ≤ j ≤ n − 2 and xj 7→ 1 for n − 1 ≤ j ≤ n. This is the saturation of I ∩ K[x0, x1,...,xn−1]. The following lemma generalizes Lemma 2.12 and Proposition 2.13 to strongly stable ideals. Lemma 3.4. Let I be a saturated strongly stable ideal. ′ (i) If I is any expansion of I, then we have pI′ = A(pI ). (ii) We have p∇(I) = ∇(pI ). j (iii) There exists j ∈ N such that pΦ(I) = A Φ(pI ). Proof. (i) Let I′ be the expansion of I at g. For all d ≥ deg g, Lemma 3.2(ii) shows that the only ′ d−deg g ′ ′ monomial in Id \ Id is gxn , so that hI (d)=1+ hI (d). Hence, we have pI =1+ pI . (ii) Let S = K[x0, x1,...,xn] and J = I ∩ K[x0, x1,...,xn−1]. Because I is saturated, xn is not a zero-divisor and
0 −→ (S/I)(−1) −→ S/I −→ K[x0, x1,...,xn−1]/J −→ 0
is a short exact sequence. The Hilbert function of J now satisfies hJ (i)= hI (i)−hI (i−1), for all i ∈ Z. Hence, by saturating J with respect to hx0, x1,...,xn−1i, we find that p∇(I)(t)= pI (t) − pI (t − 1) = [∇(pI )](t). (iii) The ideal Φ(I) is saturated strongly stable by Lemma 3.2(i),(iii) with no minimal mono- mial generators divisible by xn. Thus, ∇ Φ(I) = I. Part (ii) shows that ∇(pΦ(I))= pI , so that Φ∇(p )=Φ(p ). Lemma 2.5(iii) then shows p − Φ∇(p ) ∈ N. Φ(I) I Φ(I) Φ(I) The heart of Reeves’ algorithm [Ree92] is the following. Theorem 3.5. If I is a saturated strongly stable ideal of codimension c, then there is a finite sequence I(0),I(1),...,I(i) such that I(0) = h1i = K[x0, x1,...,xc], I(i) = I, and I(j) is an expansion or extension of I(j−1), for all 1 ≤ j ≤ i. Proof. See [Moo12, Theorem 3.20] or [MN14, Theorem 4.4]. 10 Table 1. Summary of Basic Operations
A(p)=1+ p with partitions (b , b ,...,b , 0) and Hilbert polynomial p with partitions 1 2 r (e0 +1, e1, e2,...,ed) (b1, b2,...,br) and (e0, e1,...,ed) Φ(p) with partitions (b1 +1, b2 +1,...,br +1) and (e0, e0, e1,...,ed) p A(p) p A(Ln)= Ln = L(a0 +1, a1, a2,...,an−1) lex ideal Ln = L(a0, a1,...,an−1) p Φ(p) Φ(Ln)= Ln+1 = L(0, a0, a1,...,an−1) ′ expansion I of I with pI′ = A(p) saturated str. st. ideal I with pI = p extension Φ(I)= I · K[x0, x1,...,xn+1]
When I =6 h1i, the first step is always to expand h1i to hx0, x1,...,xc−1i. We will start after this step, as we assume p =6 0. The sequences of expansions and extensions are not generally unique, but Theorem 2.14 shows that they are for lexicographic ideals. Theorem 3.5 leads to the following algorithm. The original is in [Ree92, Appendix A], but we follow [Moo12, MN14]; see also [CLMR11, Section 5]. Algorithm 3.6. Input: an admissible Hilbert polynomial p ∈ Q[t] and n ∈ N satisfying n> deg p Output: all saturated strongly stable ideals with Hilbert polynomial p in K[x0, x1,...,xn] j = 0; d = deg p; d d−1 1 q0 = ∇ (p); q1 = ∇ (p); . . . ; qd−1 = ∇ (p); qd = p; S = {hx0, x1,...,xn−d−1i}, where hx0, x1,...,xn−d−1i ⊂ K[x0, x1,...,xn−d]; WHILE j ≤ d DO T = ∅; FOR J ∈ S, considered as an ideal in K[x0, x1,...,xn−d+j] DO IF qj − pJ ≥ 0 THEN compute all sequences of qj − pJ expansions that begin with J; T = T ∪ the resulting set of sat. str. st. ideals with Hilbert polynomial qj; S = T ; j = j + 1; RETURN S Proof. See [Moo12, Algorithm 3.22]. Here, S is reset at the jth step to Moore’s S(d−j). Example 3.7. To compute all codimension 2 saturated strongly stable ideals with Hilbert polynomial p(t)=3t + 1, we first compute ∇(p) = 3. We produce all length 2 sequences of expansions beginning at hx0, x1i ⊂ K[x0, x1, x2]. The only expandable generator of hx0, x1i 2 2 2 is x1, with expansion hx0, x1i. Both x0 and x1 are expandable in hx0, x1i, with expan- 2 2 3 sions hx0, x0x1, x1i, hx0, x1i ⊂ K[x0, x1, x2]. Extending each of these to K[x0, x1, x2, x3], their Hilbert polynomials are 3t +1 and 3t, respectively. Thus, we make all possible expansions 3 2 3 3 4 3 of hx0, x1i; expansion at x0 gives hx0, x0x1, x0x2, x1i, and expansion at x1 gives hx0, x1, x1x2i. Hence, there are three codimension 2 saturated strongly stable ideals with Hilbert polynomial 2 2 2 3 4 3 3t + 1 namely, hx0, x0x1, x1i, hx0, x0x1, x0x2, x1i, and hx0, x1, x1x2i in K[x0, x1, x2, x3]. 11 4. K-Polynomials and Climbing Trees The goal of this section is to understand where Hilbert functions and Hilbert polynomials of saturated strongly stable ideals coincide. Theorem 4.6 states that among the saturated strongly stable ideals with a fixed codimension and Hilbert polynomial, the degree of the K-polynomial of the lexicographic ideal is strictly the largest. The proof tracks the gen- esis of minimal monomial generators as Algorithm 3.6 traces the path from 1 to p in M. Proposition 4.4 identifies where the inequality first occurs and Proposition 4.5 shows that it persists. The Hilbert series of a finitely generated graded K[x0, x1,...,xn]-module M is the for- i −1 mal power series HM (T )= i∈Z hM (i) T ∈ Z[T ][[T ]]. The Hilbert series of M is a rational −n−1 function HM (T ) = (1 − T ) KM (T ) and the K-polynomial of M is the numerator KM , P possibly divisible by 1 − T , of HM ; see [MS05b, Theorem 8.20]. For a quotient by a homoge- neous ideal I, we use the notation HI = HK[x0,x1,...,xn]/I and KI = KK[x0,x1,...,xn]/I . We consider a fundamental example.
−n−1 Example 4.1. If S = K[x0, x1,...,xn], then we have KS(T )= 1, as HS(T ) = (1 − T ) . −n−1 d d If d ∈ N, then we have HS(−d)(T )=(1 − T ) T and KS(−d)(T )= T .
The following well-known lemma is useful. As before, G(I) denotes the minimal set of monomial generators of a monomial ideal I.
Lemma 4.2. Let I ⊆ K[x0, x1,...,xn] be a strongly stable ideal. deg g max g (i) We have KI (T )=1 − g∈G(I) T (1 − T ) . (ii) We have deg K ≤ max deg g + max g . I Pg∈G(I) Proof. (i) This follows by Lemma 3.2(ii) and Example 4.1; also see [MS05b, Proposition 2.12]. (ii) This follows immediately from (i).
Let deg HI := deg KI −n−1. The next lemma establishes that deg HI is the maximal value where hI and pI differ.
Lemma 4.3. Let I ⊆ K[x0, x1,...,xn] be a homogeneous ideal with rational Hilbert series i −n−1 HI (T )= i∈Z hI (i) T = KI (T )(1−T ) and Hilbert polynomial pI . We have hI (i)= pI (i) for all i> deg HI , while hI (i) =6 pI (i) for i = deg HI . P d k −n−1 Proof. Let KI (T )= k=0 ckT ∈ Z[T ]. Expanding (1 − T ) and gathering terms yields
P n + i n + i − 1 n + i − d H (T )= c + c + ··· + c T i. I 0 n 1 n d n i∈N X n+i n+i−1 n+i−d Thus, the Hilbert function equals hI(i)= c0 n + c1 n + ··· + cd n , for all i ∈ Z, and the Hilbert polynomial is p (t) = c t+n + c t+n−1 + ··· + c t+n−d . Following our I 0 n 1 n d n convention in Remark 2.2, the equality n+i−j = t+n−j holds if and only if i ≥ −n + j. n n t=i This implies that h (i)= p (i) whenever i ≥ −n+d = 1+deg H , proving the first statement. I I I 12 To finish, set i = d − n − 1 = deg HI and compare the value d t + n − j d−1 t + n − j t + n − d p (i)= c = c + c I j n j n d n j=0 t=i j=0 t=i t=i X X d−1 t + n − j = c + c (−1)n j n d j=0 t=i X d−1 i+n −j with the value hI (i)= j=0 cj n + cd · 0. As cd =6 0, we are finished. p In the next two propositions,P let Ln = hm1, m2,...,mn−ℓ−1i, as in Lemma 2.10. These propositions examine the behaviour of deg KI for saturated strongly stable ideals I. p Proposition 4.4. Let Ln ⊂ K[x0, x1,...,xn] be any lexicographic ideal. p (i) If m ∈ G(Ln) is a minimal monomial generator, then m is expandable if and only if m is the smallest minimal monomial generator of its degree. p (ii) Let mn−ℓ−1 denote the last minimal monomial generator of Ln. If m =6 mn−ℓ−1 is any p p ′ p other expandable generator of Ln and (Ln) is the expansion of Ln at m, then every p ′ minimal monomial generator g ∈ G((Ln) ) satisfies deg g < 1 + deg mn−ℓ−1. (iii) Moreover, in (ii), we have deg K A(p) > deg K p ′ . Ln (Ln) Proof.
(i) By inspection, if deg mj = deg mj+1 holds, then mj is not expandable. p ′ (ii) By (i), we have deg m< deg mn−ℓ−1. But the minimal generators of (Ln) are p ′ G((Ln) )= {m1, m2,...,mn−ℓ−1}\{m} ∪ mxmax m, mxmax m+1,...,mxn−1 , and deg mj is maximized at j = n − ℓ − 1, which gives the inequality. A(p) (iii) We know Ln = hm1, m2,...,mn−ℓ−2, mn−ℓ−1xn−ℓ−2, mn−ℓ−1xn−ℓ−1,...,mn−ℓ−1xn−1i, by Example 3.3. As deg mj is maximized at mn−ℓ−1, we get deg K A(p) = deg mn−ℓ−1+n. Ln Then (ii) and Lemma 4.2(ii) yield the desired inequality. Proposition 4.5 explains how the K-polynomial inequality in Proposition 4.4 persists.
Proposition 4.5. Let I ⊂ K[x0, x1,...,xn] be a saturated strongly stable ideal, p = pI , and p mn−ℓ−1 be the last minimal generator of Ln. Consider the following condition on I:
(⋆) all generators g ∈ G(I) satisfy deg g < deg mn−ℓ−1 and max g ≤ max mn−ℓ−1. If I satisfies (⋆), then the following are true:
p p (i) deg KLn > deg KI , or equivalently, deg HLn > deg HI ; ′ ′ A(p) (ii) if I denotes any expansion of I, then I satisfies (⋆) with respect to Ln ; pΦ(I) (iii) the extension Φ(I) satisfies (⋆) with respect to Ln+1 ; and (iv) if I(0),I(1),...,I(i) is any finite sequence such that I(0) = I and I(j) is an expansion or extension of I(j−1), for all 0 < j ≤ i, then I(i) satisfies (⋆). Proof. (i) This follows immediately from Lemma 4.2(ii) and (⋆). (ii) The condition (⋆) for the expansion I′ becomes: every generator g′ ∈ G(I′) satisfies ′ ′ deg g < 1 + deg mn−ℓ−1 and max g ≤ n − 1. Both inequalities hold, by definition of the minimal monomial generators of I′ and because I satisfies (⋆). 13 Φ(p) Φ(p) (iii) An analogous condition to (⋆) holds between Φ(I) and Ln . Replacing Ln by Aj Φ(p) Ln , with j defined by Lemma 3.4(iii), results in higher degree and maximum index Aj Φ(p) of the last minimal generator of Ln ; cf. Example 3.3. Hence, Φ(I) satisfies (⋆). (iv) We apply induction to i. The case i = 1 is resolved by (ii) and (iii). If i> 1, then (ii) and (iii) ensure that I(1) satisfies (⋆), and we apply the induction hypothesis. Combining Propositions 4.4 and 4.5 leads to the main result of this section.
Theorem 4.6. Let I ⊂ K[x0, x1,...,xn] be a saturated strongly stable ideal and denote p p p p = pI . If I =6 Ln, then we have deg KLn > deg KI , or equivalently, deg HLn > deg HI . p Proof. Both I and Ln are generated by Algorithm 3.6. Let c be their codimension and I(1),I(2),...,I(i) be a finite sequence such that I(1) = hx0, x1,...,xc−1i ⊂ K[x0, x1,...,xc], I(i) = I, and I(j) is an expansion or extension of I(j−1), for all 1 < j ≤ i. Theorem 2.14 p implies that if I =6 Ln, then there is some 2 ≤ k ≤ i such that I(j) is lexicographic, for all 1 ≤ j ≤ k − 1, but I(k) is not. By Proposition 4.4(i), I(k) is the expansion of I(k−1) at a minimal generator of nonmaximal degree. Proposition 4.4(ii) then shows that I(k) satisfies (⋆). Applying Proposition 4.5(iv) to the subsequence I(k),I(k+1),...,I(i) shows that I(i) = I satisfies (⋆), hence, applying Proposition 4.5(i) finishes the proof. Proof of Theorem 1.3. Theorem 4.6 proves the claim.
Example 4.7. Example 3.7 shows the saturated strongly stable ideals in K[x0, x1, x2, x3] with 2 2 2 3 3t+1 4 3 Hilbert polynomial 3t +1 are hx0, x0x1, x1i, hx0, x0x1, x0x2, x1i, and L3 = hx0, x1, x1x2i. Lemma 4.2(i) yields deg K 3t+1 = 6, deg K 2 3 = 3, and deg K 2 2 = 3. L3 hx0,x0x1,x0x2,x1i hx0,x0x1,x1i
Corollary 4.8. Let I ⊂ K[x0, x1,...,xn] be a saturated strongly stable ideal and let p = pI . p p If I =6 Ln, then there exists k ∈ Z such that hI (j)= p(j), for all j ≥ k, but hLn (k) =6 p(k).
p Proof. Lemma 4.3 and Theorem 4.6 show that this is the case for k = deg HLn . 5. The Ubiquity of Smooth Hilbert Schemes The goal now is to investigate our proposed geography of Hilbert schemes, formally de- scribed by the collection of trees H, by applying the Hilbert series inequalities of Theo- rem 4.6. Surprisingly, we recover a not-so-well-known family of irreducible Hilbert schemes, first discovered by Gotzmann [Got89, Proposition 1]. Moreover, we observe that a complete classification of Hilbert schemes with unique strongly stable ideals can be given by examin- ing how Reeves’ algorithm interacts with Hc. These Hilbert schemes are nonsingular and irreducible over algebraically closed or characteristic 0 fields, and under natural probability distributions on the trees Hc occur with probability at least 0.5. The next two lemmas are used to prove the classification Theorem 5.3.
Lemma 5.1. Let I ⊂ K[x0, x1,...,xn] be a homogeneous ideal, and let p = pI .
p p (i) We have hI (i) ≥ hLn (i), for all i ∈ Z, where Ln is the corresponding lexicographic ideal. (ii) The Hilbert function of Φ(I)= I·K[x0, x1,...,xn+1] is given by hΦ(I)(i)= 0≤j≤i hI (j). Proof. P h (i) Section 2.2 defines the (possibly unsaturated) lexicographic ideal Ln ⊂ K[x0, x1,...,xn] for the Hilbert function h = hI . We have h p h(i) = dimK K[x0, x1,...,xn]i/(Ln)i ≥ dimK K[x0, x1,...,xn]i/(Ln)i, 14 p h p for all i ∈ Z, because Ln contains Ln. It follows that h(i) ≥ hLn (i). (ii) The homogeneous piece (Φ(I))i has decomposition i−j K i−j K Φ(I) i = Ij · xn+1 ⊂ [x0, x1,...,xn]j · xn+1 = [x0, x1,...,xn+1]i j∈N,j≤i j∈N,j≤i M M and the desired equality follows directly. Lemma 5.2. Let c > 0, p be an admissible Hilbert polynomial, and Λ be a finite sequence Λ(p) of Φ’s and A’s. The number of expandable minimal monomial generators of Lc+deg Λ(p) is p greater than or equal to the corresponding number for Lc+deg p. Proof. This follows from Proposition 4.4(i) and the definition of expandable. We can now prove our classification result. Theorem 5.3. Let p(t) = r t+bj −j+1 , for b ≥ b ≥···≥ b ≥ 0. The lexicographic j=1 bj 1 2 r ideal is the unique saturated strongly stable ideal of codimension c with Hilbert polynomial p P if and only if at least one of the following holds: (i) br > 0, (ii) c ≥ 2 and r ≤ 2, (iii) c =1 and b1 = br, or (iv) c =1 and r − s ≤ 2, where b1 = b2 = ··· = bs > bs+1 ≥···≥ br.
br br−1−br b2−b3 b1−b2 Proof. Let br > 0. Proposition 2.6 gives p = Φ AΦ ··· AΦ AΦ (1), so there exists q such that p = Φ(q). Saturated strongly stable ideals are generated by Algorithm 3.6. The procedure is recursive and generates the codimension c saturated strongly stable ideals with Hilbert polynomial p by extending all codimension c saturated strongly stable ideals with Hilbert polynomial q = ∇Φ(q) and keeping those with Hilbert polynomial p. q p By Proposition 2.13, we have Φ(Ln)= Ln+1 ⊂ K[x0, x1,...,xn+1], where n = c + deg q. It suffices to prove the following statement:
If J ⊂ K[x0,...,xn] is a saturated, strongly stable, nonlexicographic ideal, then pΦ(J) =6 Φ(pJ ).
Let I = Φ(J) be the extension of such an ideal, q = pJ , and p = Φ(q). By Lemma 3.4(iii), we must show that p − p > 0. Setting dq = deg H q , we show that h (i) > h p (i), for all I Ln I Ln+1 integers i ≥ dq. Lemma 5.1(ii) implies that
hI (i)= hJ (j)= hJ (j)+ hJ (j) and 0≤j≤i X 0≤Xj≤dq dqX
Remark 5.4. The case br > 0 is a consequence of Theorem 4.6. Another approach might exist using Stanley decompositions; see [MS05a, SW91, Sta82]. Indeed, Proposition 2.13 follows by considering a Stanley decomposition of the lexicographic ideal, while Lemma 3.2(ii) gives a Stanley decomposition of I. We thank D. Maclagan for pointing this out. A non-standard Borel-fixed ideal is a Borel-fixed ideal that is not strongly stable. Such ideals exist only in positive characteristic. Pardue [Par94, Chapter 2] gives the following combinatorial criterion for I ⊂ K[x0, x1,...,xn] to be Borel-fixed when K is infinite of char- ℓ acteristic p > 0: I is monomial and for all monomials m ∈ I, if xjkm and xi ≻ xj, then −k k ℓ ℓ ℓ+1 xj xi m ∈ I holds, for all k ≤p ℓ. Here, xjkm means xj divides m but xj does not; k ≤p ℓ i i means that in the base-p expansions k = i kip ,ℓ = i ℓip , we have ki ≤ ℓi, for all i; also see [Eis95, § 15.9.3]. When K has characteristic 0, Theorem P5.3 generalizesP a result of Gotzmann. In fact, the Hilbert schemes with br > 0 are the irreducible ones in [Got89]. These include Grassmannians and the Hilbert schemes of hypersurfaces studied in [Adl85˚ ]. We extend our classification to positive characteristic. Corollary 5.5. Let K be an algebraically closed field and p,c be as in Theorem 5.3. Then Hilbp(Pn) has a unique Borel-fixed point, where n = c + deg p. Proof. Let I be the saturated Borel-fixed ideal of a point on Hilbp(Pn). So I is monomial and no g ∈ G(I) is divisible by xn, by [Par94, II, Proposition 9]. Suppose br > 0. As xn is not a zero-divisor of K[x0, x1,...,xn]/I, the result [Got89, Proposition 2] applies directly, showing that I = hf0x0, f0f1x1, f0f1f2x2,...,f0f1 ··· fk−1xk−1, f0f1 ··· fki, where fi ∈ K[xi,...,xn] is homogeneous and k ≤ n − 2; cf. [RS97, Theorem 4.1]. We may assume each fi is monomial, of degree di. Then Pardue’s criterion shows that I contains the saturated lexicographic ideal
d0+1 d0 d1+1 d0 d1 d2+1 d0 d1 dk−1+1 d0 d1 dk L := hx0 , x0 x1 , x0 x1 x2 ,...,x0 x1 ··· xk−1 , x0 x1 ··· xk i. p The degrees di determine the Hilbert polynomials of I and L, so we must have I = L = Ln. 16 Let c ≥ 2 and br = 0. The cases p = 1 and p = 2 follow by inspection. When r = 2 and b1 > 0, consider the ideal J := I ∩ K[x0, x1,...,xn−1], which is Borel-fixed, has Hilbert polynomial ∇(p), and satisfies Φ(J) = I as I is saturated. By the previous cases, we know ∞ ∇(p) ∞ p−1 that (J : xn−1) = Ln−1 = hx0, x1,...,xc−1i, which further implies (I : xn−1) = Ln , by p−1 Lemma 3.4(iii), Proposition 2.13, and [Par94, II, Proposition 9]. So I2 ⊂ (Ln )2 is a codi- 2 2 mension 1 vector subspace, the latter being spanned by x0, x0x1,...,xc−1, xc−1xc,...,xc−1xn. Now Pardue’s criterion implies that we can only obtain I2 by removing xc−1xn, i.e. I is the p−1 lex-expansion of Ln , as I is generated in degree r = 2. Now let c = 1 and br = 0. If b1 = br, then I is lexicographic, as it is monomial and saturated. If b1 > br, then the argument is entirely analogous to the previous paragraph. Proof of Theorem 1.1. Theorem 5.3 and Corollary 5.5 prove the claim. Geometrically, the Hilbert schemes corresponding to Theorem 5.3 are well-behaved. Lemma 5.6. Let K be an infinite field. If the lexicographic ideal is the unique saturated Borel-fixed ideal with Hilbert polynomial p and codimension c, then Hilbp(Pn) is nonsingular and irreducible, where n = c + deg p.
p n Proof. Every component and intersection of components of Hilb (P ) contains a point [XI ] defined by a saturated Borel-fixed ideal I; the Remarks in [Ree95, § 2] hold over infinite fields, by [BS87b, Proposition 1] and [Eis95, Theorem 15.17]. Lexicographic points are nonsingu- p lar by [RS97, Theorem 1.4], so XLn cannot lie on an intersection of components. Thus, Hilbp(Pn) has a unique, generically nonsingular, irreducible component. p n Suppose Hilb (P ) has a singular point, given by I ⊂ K[x0, x1,...,xn]. For γ ∈ GLn+1(K), the point [Xγ·I ] is also singular, and for generic γ ∈ GLn+1(K), the initial ideal of γ · I with respect to any monomial ordering is Borel-fixed, by Galligo’s Theorem [Gal74, BS87b]. Thus, a one-parameter family of singular points degenerating to the lexicographic ideal exists. By upper semicontinuity of cohomology of the normal sheaf, the lexicographic ideal is singular, a contradiction; see [Har77, III, Theorem 12.8], [Har10, Theorem 1.1(b)]. Hence, Hilbp(Pn) is nonsingular and irreducible. Thus, our classification provides irreducible, nonsingular Hilbert schemes over algebraically closed or characteristic 0 fields. Moreover, these Hilbert schemes are rational, by [LR11, Theorem C]. We now wish to make a quantitative statement about the prevalence of this behaviour. To do so, we need a probability measure Pr: 2H → [0, 1], which is determined by a normalized nonnegative function pr: H → R, as in [Bil95, Examples 2.8–2.9]. That is, for every subset N H N ⊆ , we have Pr( ) := H∈N pr(H). However, there is no known canonical distribution on H. A natural choice on Hc is to mimic uniform distribution by making all vertices at a P −k fixed height equally likely; given a mass function fc : N → [0, 1], let prc(H)=2 fc(k), for all H ∈ Hc at height k. Distributions on H are then specified via functions fc : N → [0, 1], −k for all Hc, and a mass function f : N \{0} → [0, 1], by setting pr(H)=2 f(c)fc(k) (the −k−1 −c probability of H), for all H ∈ Hc at height k. Using fc(k) := 2 and f(c) := 2 is sufficient for us, although other basic examples (geometric, Poisson, etc.) work similarly. Theorem 5.7. Let K be an algebraically closed or characteristic 0 field and endow H with the structure of a probability space as in the preceding paragraph. The probability that a random Hilbert scheme is irreducible and nonsingular is greater than 0.5. 17 Proof. Let N be the set of nonsingular and irreducible Hilbert schemes. We compute 2k f(c)f (k) Pr(N )= pr (H) ≥ f(c)f (0) + c c 2 2k N c>0 ! HX∈ X Xk≥1 f(c)f (0) f(c)f (k) = c + c 2 2 c>0 N ! 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