The Ubiquity of Smooth Hilbert Schemes

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by  Andrew P. Staal

THE UBIQUITY OF SMOOTH HILBERT SCHEMES
ANDREWP.STAAL
7
1 ABSTRACT. We investigate the geography of Hilbert schemes that parametrize closed sub-
0 schemesofprojectivespacewithaspecifiedHilbertpolynomial. WeclassifyHilbertschemes
2 with unique Borel-fixed points via combinatorial expressions for their Hilbert polynomials.

n We realize the set of all nonempty Hilbert schemes as a probability space and prove that
a Hilbertschemesareirreducibleandnonsingular withprobabilitygreaterthan0.5.
J

1
3 1. INTRODUCTION

] Hilbert schemes parametrizing closed subschemes with a fixed Hilbert polynomial in
G
projective space are fundamental moduli spaces. With the exception of Hilbert schemes
A
parametrizing hypersurfaces [ACG11, Example 2.3] and points in the plane [Fog68], the
.
h geometric features of typical Hilbert schemes are still poorly understood. Techniques for
t
a producingpathologicalHilbertschemes areknown,generatingHilbertschemes withmany
m
irreduciblecomponents[Iar72,FP96],withgenericallynonreducedcomponents[Mum62],
[ andwitharbitrarysingularitytypes[Vak06]. WhatshouldweexpectfromarandomHilbert

1 scheme? Can we understand the geography of Hilbert schemes? Our answer is that the
v
set of nonempty Hilbert schemes forms a graph and a discrete probability space, and that
0
8 irreducible nonsingular Hilbert schemes are unexpectedly common.
0 Let Hilbp(Pn) denote the Hilbert scheme parametrizing closed subschemes of Pn with
0
Hilbert polynomial p. Polynomials that are Hilbert polynomials of homogeneous ideals are
0
. classified in [Mac27]. Any such admissible Hilbert polynomial p(t) has a combinatorial
2
0 expression r t+bj−(j−1) for b b b 0. Our first main result is the
7 j=1 bj 1 ≥ 2 ≥ ··· ≥ r ≥
following.
1
P (cid:0) (cid:1)
:
v Theorem 1.1. The lexicographic ideal is the unique saturated Borel ideal of codimension c
i
X with Hilbert polynomial p if and only if:
r
a (i) c 2 and either br > 0 or r 2; or
≥ ≤
(ii) c = 1andeitherb > 0,b = b ,orr s 2,whereb = b = = b > b b .
r 1 r 1 2 s s+1 r
− ≤ ··· ≥ ··· ≥
Borel ideals generalize lexicographic ideals and in characteristic 0 define Borel-fixed
points on Hilbert schemes. Some basic general properties of Hilbert schemes have been
extracted from these ideals. Rational curves linking Borel-fixed points prove connected-
nessin [Har66, PS05]. The thesis [Bay82] uses them to give equationsforHilbertschemes
and proposes studying their tangent cones. Further, [Ree95] studies their combinatorial
properties to give general bounds for radii of Hilbert schemes, and [RS97] proves that
lexicographic points are nonsingular. Theorem 1.1 specifies an explicit collection of well-
behavedHilbert schemes, generalizesthe main resultof [Got89], andimproves ourunder-
standing of the geography of Hilbert schemes.
This collection of Hilbert schemes is ubiquitous. Our new interpretation of Macaulay’s
classification identifies an infinite binary tree H whose vertices are the Hilbert schemes
c
1
Hilbp(Pn) parametrizing codimension c = n degp subschemes, for each positive c Z.
− ∈
Assuming that vertices at a fixed height are equally likely, combining probability distribu-
tions for the height with a distribution for the parametrized codimension c endows the
set of Hilbert schemes with the structure of a discrete probability space. This leads to our
second main result.
Theorem 1.2. 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 gen-
erating saturated Borel ideals first described in [Ree92] and later generalized in [Moo12,
CLMR11]. We obtain precise information about Hilbert series and K-polynomials of satu-
rated Borel ideals. The primary technical result we need is the following.
Theorem 1.3. Let I K[x ,x ,...,x ] be a saturated Borel ideal with Hilbert polynomial p,
0 1 n
⊂
letLp bethecorrespondinglexicographicidealinK[x ,x ,...,x ],andletK bethenumerator
n 0 1 n I
of the Hilbert series of I. If I 6= Lpn, then we have degKI < degKLpn.
The structure of the paper is as follows. In Section 2, we introduce two binary rela-
tions on the set of admissible Hilbert polynomials and show that they generate all such
polynomials; see Theorem 2.10. The set of lexicographic ideals is then partitioned by codi-
mension into infinitely many binary trees in Section 3. Geometrically, these are trees of
Hilbert schemes, as every Hilbert scheme contains a unique lexicographic ideal. Section 4
makes explicit this graph-theoretical structure on the set of Hilbert schemes. To identify a
sufficiently dense family of irreducible, nonsingular Hilbert schemes, we review saturated
Borel ideals in Section 5 and we examine their K-polynomials in Section 6. The main
results are in Section 7.
Conventions. Throughout,Kisanalgebraicallyclosedfield,Nisthesetofnonnegativein-
tegers,andK[x ,x ,...,x ]isthestandardZ-gradedpolynomialring. TheHilbertfunction,
0 1 n
polynomial,series,andK-polynomialofthequotientK[x ,x ,...,x ]/I byahomogeneous
0 1 n
ideal I are denoted h ,p ,H , and K , respectively.
I I I I
Acknowledgments. We especially thank Gregory G. Smith for his guidance in this re-
search. We thank Mike Roth, Ivan Dimitrov, and Tony Geramita for lessons and lectures,
and Chris Dionne, Ilia Smirnov, Nathan Grieve, Andrew Fiori, Simon Rose, and Alex Dun-
can for many discussions. 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. THE TREE OF ADMISSIBLE HILBERT POLYNOMIALS
Weidentifyagraphstructureonthesetofnumericalpolynomialsdeterminingnonempty
Hilbert schemes of projective spaces. The pioneering work [Mac27] gives a combinatorial
classificationofthese polynomials. We introduce twobinaryrelationsto equipthis setwith
the structure of an infinite binary tree.
Let K be an algebraically closed field and let K[x ,x ,...,x ] denote the homogeneous
0 1 n
(standard Z-graded) coordinate ring of n-dimensional projective space Pn. Let M be a
finitely generated graded K[x ,x ,...,x ]-module. The Hilbert function h : Z Z of M
0 1 n M
→
2
is defined by h (i) := dim (M ) for all i Z. Every such M has a Hilbert polynomial p ,
M K i ∈ M
that is, a polynomial p (t) Q[t] such that h (i) = p (i) for i 0; see [BH93, Theo-
M ∈ M M ≫
rem 4.1.3]. For a homogeneous ideal I K[x ,x ,...,x ], let h and p denote the Hilbert
⊂ 0 1 n I I
function and Hilbert polynomial of the quotient module K[x ,x ,...,x ]/I, respectively.
0 1 n
If X Pn is a nonempty closed subscheme, then there is a unique saturated homoge-
⊆
neous ideal I K[x ,x ,...,x ] such that X = Proj K[x ,x ,...,x ]/I ; see [Har77,
X 0 1 n 0 1 n X
⊆
Corollary II.5.16]. We define the Hilbert function h of X to be the Hilbert function
X
(cid:0) (cid:1)
h = h , and the Hilbert polynomial p of X to be p = p .
IX K[x0,x1,...,xn]/IX X IX K[x0,x1,...,xn]/IX
As a first example, we describe the Hilbert polynomial of Pn.
Example 2.1. Fix a nonnegative integer n N. The stars-and-bars argument [Sta12,
∈
Section 1.2] shows that the number of independent homogeneous polynomials of degree
i Z in n+1 variables is n+i and equals 0 for i < 0. That is, we have h (i) = n+i for
∈ n S n
S := K[x ,x ,...,x ]. The equality h (i) = p (i) is only valid for i n, because p only
0 1 n (cid:0) (cid:1) S S ≥ − (cid:0) S (cid:1)
has roots n, (n 1),..., 1, whereas h (i) = 0 for all i < 0.
S
− − − −
Remark 2.2. We often treat binomial coefficients as polynomials. Following [GKP94,
Section 5.1], for a variable t and a,b Z, let t+a := (t+a)(t+a−1)···(t+a−b+1) Q[t] if
∈ b b! ∈
b 0, and t+a := 0 otherwise. If b 0, then t+a has degree b in t, with zeros
≥ b ≥ (cid:0) (cid:1) b
a, (a 1),..., (a b + 1), so that t+a = j+a = 0 for j < a. Interestingly,
− − − (cid:0) (cid:1)− − b |t=j 6 (cid:0)b (cid:1) −
[Mac27, p. 533] uses distinct notation for polynomial and integer binomial coefficients.
(cid:0) (cid:1) (cid:0) (cid:1)
A polynomial is an admissible Hilbert polynomial if it is the Hilbert polynomial of a
nonempty closed subscheme of a projective space. As a consequence, admissible Hilbert
polynomialscorrespondtononemptyHilbertschemes. Wehavethefollowingclassification.
Proposition 2.3. The following conditions are equivalent:
(i) The polynomial p(t) Q[t] is an admissible Hilbert polynomial.
∈
(ii) There exist integers e e e > 0 such that p(t) = d t+i t+i−ei .
0 ≥ 1 ≥ ··· ≥ d i=0 i+1 − i+1
(iii) There exist integers b b b 0 such that p(t) = r t+bj−(j−1) .
1 ≥ 2 ≥ ··· ≥ r ≥ Pj=1(cid:0) (cid:1)bj (cid:0) (cid:1)
Moreover, the correspondences between admissible Hilbert polynomials and sequences of e ’s,
P (cid:0) (cid:1) i
or of b ’s, are bijective.
j
Proof.
(i) (ii) This is proved in [Mac27, Part I]; see the formula for “χ(ℓ)” at the bottom of
⇔
p. 536. For a modern 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].
⇔
The uniqueness of the sequences of integers attached to an admissible polynomial is also
explained by the aforementioned sources. (cid:3)
We define the Macaulay–Hartshorne expression of an admissible Hilbert polynomial p
to be its expression p(t) = d t+i t+i−ei , for e e e > 0. Similarly,
i=0 i+1 − i+1 0 ≥ 1 ≥ ··· ≥ d
we define the Gotzmann expression of p to be its expression p(t) = r t+bj−(j−1) , for
P (cid:0) (cid:1) (cid:0) (cid:1) j=1 bj
b b b 0. From these, we see the degree d = b , the leading coefficient e /d!,
1 ≥ 2 ≥ ··· ≥ r ≥ 1 P (cid:0) (cid:1)d
and the Gotzmann number r of p, which bounds the Castelnuovo–Mumford regularity of
saturated ideals with Hilbert polynomial p; see [IK99, Definition C.12].
Macaulay–HartshorneandGotzmannexpressionsareconjugate. Theconjugatepartition
to a partition λ := (λ ,λ ,...,λ ) of an integer ℓ = k λ is the partition of ℓ obtained
1 2 k i=1 i
3
P
from the Ferrers diagram of λ by interchanging rows and columns, having λ λ parts
i i+1
−
equal to i; see [Sta12, Section 1.8].
Lemma 2.4. If p(t) Q[t] is an admissible Hilbert polynomial with Macaulay–Hartshorne
∈
expression d t+i t+i−ei , for e e e > 0, and Gotzmann expression
i=0 i+1 − i+1 0 ≥ 1 ≥ ··· ≥ d
r t+bj−(j−1) , for b b b 0, then r = e and the nonnegative partition
j=1 bPj (cid:0) (cid:1) 1 (cid:0)≥ 2 ≥(cid:1) ··· ≥ r ≥ 0
(b ,b ,...,b ) is conjugate to the partition (e ,e ,...,e ).
P1 2(cid:0) r (cid:1) 1 2 d
Proof. Rewriting the Macaulay–Hartshorne expression of p as
d d−1
t+i t+i e t+i e t+i e
d d i
− + − − ,
i+1 − i+1 i+1 − i+1
i=0 (cid:18) (cid:19) (cid:18) (cid:19) i=0 (cid:18) (cid:19) (cid:18) (cid:19)
X X
weprovethat d t+i t+i−ed = ed t+d−(j−1) . Ifd = 0, then t t−e0 = e holds.
i=0 i+1 − i+1 j=1 d 1 − 1 0
If d > 0, thenPwe h(cid:0)ave(cid:1) di(cid:0)=0 it++1i(cid:1) −Pt+ii+−1ed(cid:0) = di(cid:1)=−01 it++1i − t+ii+−1e(cid:0)d (cid:1) +(cid:0) dt++d1(cid:1) − t+dd+−1ed .
By induction, this equalPs (cid:0)ed (cid:1)t+(d(cid:0)−1)−(j−(cid:1)1) h+Pt+d(cid:0) (cid:1)t+d−(cid:0)ed . T(cid:1)hie a(cid:0)ddit(cid:1)ion(cid:0)formul(cid:1)a
j=1 d−1 d+1 − d+1
[GKP94, Section 5.1] yieldhs i
P (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
t+d t+d e ed t+d (j 1) t+d e
− d = − − + − d and
d+1 − d+1 d d
" #
(cid:18) (cid:19) (cid:18) (cid:19) j=2 (cid:18) (cid:19) (cid:18) (cid:19)
X
t+d ed t+(d 1) (j 1) t+d e
d
= − − − + − ,
d d 1 d
" #
(cid:18) (cid:19) j=1 (cid:18) − (cid:19) (cid:18) (cid:19)
X
and we obtain d t+i t+i−ed = ed t+d−(j−1) , as desired.
i=0 i+1 − i+1 j=1 d
Now we write the Macaulay–Hartshorne expression as
P (cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1)
ed t+d (j 1) d−1 s+i s+i (e e )
p(t) = − − + − i − d
d i+1 − i+1
" #
Xj=1 (cid:18) (cid:19) Xi=0 (cid:18) (cid:19) (cid:18) (cid:19) s:=t−ed
and repeat the decomposition on the second part. With s := t e , this yields the sum
d
−
d−1 d−2
s+i s+i (e e ) s+i (e e ) s+i (e e )
d−1 d d−1 d i d
− − + − − − − ,
i+1 − i+1 i+1 − i+1
" #
i=0 (cid:18) (cid:19) (cid:18) (cid:19) i=0 (cid:18) (cid:19) (cid:18) (cid:19)
X X
whosefirstpartequals ed−1−ed s+(d−1)−(k−1) ,bythepreviousparagraph. Reindexingwith
k=1 d−1
j := k +e and evaluating at s := t e gives ed−1 t+(d−1)−(j−1) . Therefore, we have
d P (cid:0) − d (cid:1) j=ed+1 d−1
d t+i t+i−ed = r t+bj−(j−1) when b = i for all e j > e , where e := 0.
i=0 i+1 − i+1 j=1 bj Pj (cid:0) i ≥ (cid:1) i+1 d+1
This shows that r = e and that e e parts equal i in the partition associated to the
P (cid:0) (cid:1) (cid:0) (cid:1) 0P (cid:0) i − (cid:1)i+1
Gotzmann expression of p, for all 0 i d. Finally, r b = d (e e )i = d e
≤ ≤ j=1 j i=0 i − i+1 i=1 i
holds and it follows that (b ,b ,...,b ) is conjugate to (e ,e ,...,e ). (cid:3)
1 2 r 1 2 d
P P P
We define the Macaulay–Hartshorne partition of an admissible Hilbert polynomial
p(t) = d t+i t+i−ei to be the partition (e ,e ,...,e ), and the Gotzmann parti-
i=0 i+1 − i+1 0 1 d
tion of p(t) = r t+bj−(j−1) to be the nonnegative partition (b ,b ,...,b ).
P (cid:0) j(cid:1)=1 (cid:0) bj (cid:1) 1 2 r
4
P (cid:0) (cid:1)
Example 2.5. The twisted cubic curve X P3 is defined by the ideal I K[x ,x ,x ,x ]
X 0 1 2 3
⊂ ⊂
of 2-minors of the matrix [x0 x1 x2]. The Macaulay–Hartshorne and Gotzmann expressions
x1 x2 x3
for the Hilbert polynomial of the twisted cubic are
t+0 t+0 4 t+1 t+1 3
p (t) = 3t+1 = − + −
X 0+1 − 0+1 1+1 − 1+1
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
t+1 t+1 1 t+1 2 t+0 3
= + − + − + − ,
1 1 1 0
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
respectively. The partitions are (e ,e ) = (4,3) and (b ,b ,b ,b ) = (1,1,1,0). Observe that
0 1 1 2 3 4
(3) is conjugate to (1,1,1), but that (1,1,1,0) has r = e = 4.
0
We describe two binary relations on the set of admissible Hilbert polynomials. Let p
be an admissible Hilbert polynomial with Macaulay–Hartshorne partition (e ,e ,...,e )
0 1 d
and Gotzmann partition (b ,b ,...,b ). We define a mapping Φ, from the set of admissible
1 2 r
Hilbertpolynomialstoitself,thattakesptothepolynomialΦ(p)withMacaulay–Hartshorne
partition (e ,e ,e ,...,e ) and Gotzmann partition (b +1,b +1,...,b +1). Explicitly, we
0 0 1 d 1 2 r
have [Φ(p)](t) = t+0 t+0−e0 + d+1 t+i t+i−ei−1 = r t+(bj+1)−(j−1) , which is
0+1 − 0+1 i=1 i+1 − i+1 j=1 bj+1
admissible, by Proposition 2.3.
(cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1)
To define the second binary relation on the set of admissible Hilbert polynomials, let
Ψ: Q[t] Q[t]bethemappingtakingapolynomialpto1+p. Ifp(t)isanadmissibleHilbert
→
polynomial with Macaulay–Hartshorne partition (e ,e ,...,e ) and Gotzmann partition
0 1 d
(b ,b ,...,b ), then Ψ(p) = 1+p is also an admissible Hilbert polynomial, with Macaulay–
1 2 r
Hartshorne partition (e + 1,e ,e ,...,e ) and Gotzmann partition (b ,b ,...,b ,0). We
0 1 2 d 1 2 r
have [Ψ(p)](t) = t+0 t+0−(e0+1) + d t+i t+i−ei = r t+bj−(j−1) + t+0−r .
0+1 − 0+1 i=1 i+1 − i+1 j=1 bj 0
Therefore, the restriction of Ψ defines a mapping from the set of admissible Hilbert poly-
(cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1) (cid:0) (cid:1)
nomials to itself, which we also denote by Ψ.
Example 2.6. The simplest admissible Hilbert polynomial is that of a reduced point in
projective space, with Gotzmann expression 1 = t+0−0 . We have Φ(1) = t+1−0 = t+1,
0 1
which is the Hilbert polynomial of a reduced line. Also, Ψ(1) = t+0−0 + t+0−1 = 2 is the
(cid:0) (cid:1) 0 (cid:0)0 (cid:1)
Hilbert polynomial of two points in projective space.
(cid:0) (cid:1) (cid:0) (cid:1)
TobetterunderstandΦ,weconsiderthebackwardsdifferenceoperator : Q[t] Q[t],
∇ →
definedbyq [ (q)](t) := q(t) q(t 1); comparewith[BH93, Lemma4.1.2]. We collect
7→ ∇ − −
elementary properties showing the interplay between Ψ, Φ, and . Backwards differences
∇
arediscretederivatives,andLemma2.7(ii)showsthatΦgivesindefinitesumsofadmissible
polynomials. Part (iii) is a well-known discrete analogue of the Fundamental Theorem of
Calculus.
Lemma 2.7. If p(t) is an admissible Hilbert polynomial with Macaulay–Hartshorne partition
(e ,e ,...,e ) and Gotzmann partition (b ,b ,...,b ), then the following hold:
0 1 d 1 2 r
(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) ΨaΦ(p) = p, for all a N;
∇ P (cid:0) ∈ (cid:1) P (cid:0) (cid:1) (cid:0) (cid:1)
(iii) if degp > 0 and k 1,2,...,r is the largest index such that b = 0, then we have
k
∈ { } 6
p Φ (p) = r k, but if degp = 0, then (p) = 0; and
− ∇ − ∇
(iv) [(ΦΨ ΨΦ)(p)](t) = t r.
− −
5
Remark 2.8. Setting a = 0 in Part (ii) shows that Φ(p) = p, so that Part (iii) shows that
∇
( Φ Φ )(p) = r k. In other words, applying Φ and then returns p, but applying
∇ − ∇ − ∇ ∇
and then Φ may alter the constant term of p.
Proof.
(i) Because is a linear operator on Q[t], it suffices to prove the statement for polyno-
∇
mials of the form t+b−i , for b,i N. By definition of and the addition formula,
b ∈ ∇
we have [ (p)](t) = t+b−i t−1+b−i = t+b−1−i .
∇ (cid:0) (cid:1)b − b b−1
(ii) Wehave[ Φ(p)](t) =(cid:0) (cid:1) r(cid:0) t+bj+(cid:1)1−(j−(cid:0)1) =(cid:1) r t+bj−(j−1) = p(t),byPart(i).
∇ ∇ j=1 bj+1 j=1 bj
Further, ΨaΦ(p) = a(cid:16)+Φ(p) , which equ(cid:17)als Φ(p) = p.
P (cid:0) (cid:1) P (cid:0) (cid:1)
∇ ∇ ∇
(iii) Because degp > 0, there is a largest index k 1,...,r such that b = 0. Thus,
k
(cid:0) (cid:1) ∈ { } 6
[Φ (p)](t) = Φ k t+bj−1−(j−1) = k t+bj−(j−1) . Hence, terms of the form
∇ j=1 bj−1 j=1 bj
t+0−(j−1) are dr(cid:16)oPpped,(cid:0)and as they(cid:1)(cid:17)all eqPual 1(cid:0), we are le(cid:1)ft with p Φ (p) = r k.
0 − ∇ −
(iv) We have [ΦΨ(p)](t) = r+1 t+bj+1−(j−1) , where b := 0. On the other hand, we
(cid:0) (cid:1) j=1 bj+1 r+1
also have [ΨΦ(p)](t) = r t+bj+1−(j−1) + t+0−r , and takingthe difference yields
Pj=1(cid:0) bj+1 (cid:1) 0
the polynomial ΦΨ ΨΦ (p) (t) = (t+1 r) 1 = t r. (cid:3)
−P (cid:0) (cid:1) −(cid:0) −(cid:1) −
Example 2.9. Examp(cid:2)le(cid:0) 2.5 shows(cid:1)tha(cid:3)t p (t) = t+1 + t + t−1 + t−3 for the twisted
X 1 1 1 0
cubic curve X P3, so we have [Φ(p )](t) = t+2 + t+1 + t + t−2 = 3t2 + 5t 1
⊂ X (cid:0)2 (cid:1) (cid:0)2(cid:1) (cid:0) 2(cid:1) (cid:0) 1 (cid:1) 2 2 −
and [ Φ(p )](t) = 3t2 + 5t 1 = 3t + 1 = p . In the other order, we find that
∇ X ∇ 2 2 − (cid:0) (cid:1)X (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
[ (p )](t) = t + t−1 + t−2 = 3 and [Φ (p )](t) = t+1 + t + t−1 = 3t.
∇ X 0 0 (cid:0) 0 (cid:1) ∇ X 1 1 1
Ifweletq(t) := 3t2+5t+1,thenwealsoobtain[ (q)](t) = 3t+1. Infact, theexpression
(cid:0) (cid:1) 2(cid:0) (cid:1)2 (cid:0) (cid:1) ∇ (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
q(t) = Ψ2Φ(p ) (t) = t+2 + t+1 + t + t−2 + t−4 + t−5 shows that the polynomial
X 2 2 2 1 0 0
q is an admissible Hilbert polynomial. This polynomial is the Hilbert polynomial of the
(cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(minimallyembedded)firstHirzebruchsurface,alsoknownastheblow-upofP2 atapoint.
The mappings Φ and Ψ endow the set of admissible Hilbert polynomials with the struc-
ture of a graph.
Theorem 2.10. The graph whose vertices correspond to admissible Hilbert polynomials and
whose edges correspond to pairs of the form p,Ψ(p) and p,Φ(p) , for all admissible Hilbert
polynomials p, forms an infinite binary tree. Moreover, the root of the tree corresponds to the
(cid:0) (cid:1) (cid:0) (cid:1)
constant polynomial 1.
The infinite binary tree has 2j vertices at height j, forall j N. We define the Macaulay
tree M to be the infinite binary tree of admissible Hilbert pol∈ynomials in Theorem 2.10.
Proof. We prove that the admissible Hilbert polynomial p(t) = d t+i t+i−ei equals
i=0 i+1 − i+1
p = Ψe0−e1ΦΨe1−e2Φ ΦΨed−1−ed ΦΨedP−1(1),(cid:0) (cid:1) (cid:0) (cid:1)
···
where e e e > 0. We proceed by induction on the length d + 1 of the
0 1 d
≥ ≥ ··· ≥
partition (e ,e , ,e ). If d = 0, then we have t+0 t+0−e0 = e = Ψe0−1(1), which
0 1 ··· d 0+1 − 0+1 0
proves the claim. The induction hypothesis on the partition (e ,e , ,e ), shows that
(cid:0) (cid:1) (cid:0) (cid:1)1 2 ··· d
d−1 t+i t+i−ei+1 = Ψe1−e2ΦΨe2−e3Φ ΦΨed−1−ed ΦΨed−1(1). Applying Φ to both
i=0 i+1 − i+1 ···
sides, we obtain e + d t+i t+i−ei = ΦΨe1−e2ΦΨe2−e3Φ ΦΨed−1−ed ΦΨed−1(1)
P (cid:0) (cid:1) (cid:0) 1 (cid:1) i=1 i+1 − i+1 ···
and applying Ψe0−e1 to both sides yields the desired equality. Because every finite binary
P (cid:0) (cid:1) (cid:0) (cid:1)
sequence of Ψ’s and Φ’s has a unique such expression, we obtain the result. (cid:3)
6
A portion of M is displayed in Figure 1, in terms of Gotzmann expressions.
0t + t−01 +
(cid:0)(cid:1) t−02(cid:0) +(cid:1)
(cid:0)0tt(cid:1)−02++(cid:0)t−01t(cid:1)−03+ (cid:0)t−0(cid:0)3(cid:1)+(cid:1)(cid:0)t−04(cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) t+1 + t +
1 1
(cid:0)t−11(cid:1)+(cid:0)t(cid:1)−12
0t + t−01 + t−02 (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0)(cid:1) (cid:0) (cid:1) (cid:0) (cid:1) t+1 + t +
1 1
(cid:0)t−11(cid:1)+(cid:0)t(cid:1)−03
t+11 + 1t + t−11 (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0)(cid:1) (cid:0) (cid:1) t+2 + t+1 + t
2 2 2
0t + t−01 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0) (cid:1) t+11 + 1t +
(cid:0)t−02(cid:1)+(cid:0)t(cid:1)−03
t+11 + 1t + t−02 (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0)(cid:1) (cid:0) (cid:1) t+2 +
2
Ψ t+2(cid:0)1 +(cid:1) t−11
t+1 + t (cid:0) (cid:1) (cid:0) (cid:1)
1 1
(cid:0) (cid:1) (cid:0)(cid:1) t+2 +
2
t+2(cid:0)1 +(cid:1) t−02
t+22 + t+21 (cid:0) (cid:1) (cid:0) (cid:1)
t (cid:0) (cid:1) (cid:0) (cid:1) t+33 + t+32
0
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:0)(cid:1)
t+11 + t−01 +
t+1 + (cid:0) t−0(cid:1)2 +(cid:0) t−0(cid:1)3
t−0(cid:0)11+(cid:1) t−02 (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) t+22 + 1t + t−11
Φ t+11 + t−01 (cid:0) (cid:1) (cid:0)(cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) t+22 + 1t + t−02
t+2 + t (cid:0) (cid:1) (cid:0)(cid:1) (cid:0) (cid:1)
2 1
(cid:0) (cid:1) (cid:0)(cid:1) t+3 + t+1
3 2
t+1 (cid:0) (cid:1) (cid:0) (cid:1)
1
t+2 +
(cid:0) (cid:1) 2
t−0(cid:0)1 +(cid:1) t−02
t+22 + t−01 (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) t+3 + t
3 1
t+2
2 (cid:0) (cid:1) (cid:0)(cid:1)
(cid:0) (cid:1) t+33 + t−01
t+3 (cid:0) (cid:1) (cid:0) (cid:1)
3
(cid:0) (cid:1) t+4
4
(cid:0) (cid:1)
FIGURE 1. The Macaulay tree M to height 4 with Gotzmann expressions
Remark 2.11. The path from the root 1 of the tree M to an admissible Hilbert polynomial
p(t) := d t+i t+i−ei = r t+bj−(j−1) isalsoencodedintheGotzmannexpression.
i=0 i+1 − i+1 j=1 bj
In particular, we have the conjugate version
P (cid:0) (cid:1) (cid:0) (cid:1) P (cid:0) (cid:1)
p = Φbr ΨΦbr−1−br Ψ ΨΦb2−b3 ΨΦb1−b2(1)
···
7
of the expression in the proof of Theorem 2.10, where b b b 0. Moreover,
1 2 r
≥ ≥ ··· ≥ ≥
these explicit expressions for the path from 1 to p(t) show that the height of the vertex p(t)
is e +d 1 = r +b 1.
0 1
− −
Example 2.12. The Hilbert polynomial of the twisted cubic curve X P3 has partitions
⊂
(e ,e ) = (4,3) and (b ,b ,b ,b ) = (1,1,1,0); see Example 2.5. The Macaulay–Hartshorne
0 1 1 2 3 4
expressionandtheproofofTheorem2.10giveΨ4−3ΦΨ3−1(1) = 3t+1,whiletheGotzmann
expression and Remark 2.11 give Φ0ΨΦ1−0ΨΦ1−1ΨΦ1−1(1) = 3t+1. This path is shown
in Figure 2. From Example 2.9, we find that the path associated to the Hilbert polynomial
of the minimally embedded Hirzebruch surface P(O O ( 1)) P4 is Ψ2ΦΨΦΨ2(1).
P1 P1
⊕ − ⊂
5
4
4t 2
−
3
Φ Ψ 3t+1
Ψ
3t
(3/2)t2+(3/2)t+1
2
2t+3
2t+2
t2+3t
Ψ 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
8
3. THE FOREST OF LEXICOGRAPHIC IDEALS
This section connects lexicographic ideals with the Macaulay tree M. Specifically, Theo-
rem 3.9 shows that M reappears infinitely many times in the set of saturated lexicographic
ideals,withexactlyonetreeL foreachpositivecodimensionc Z. Toprovethis,westudy
c
∈
two mappings on the set of lexicographic ideals, defined in analogy with Φ and Ψ. Explicit
monomial generators of lexicographic ideals given in terms of Macaulay–Hartshorne ex-
pressions help to understand Hilbert polynomials of images of lexicographic ideals under
our two mappings.
Lexicographic, or lex-segment, ideals are monomial ideals whose homogeneous pieces
are spanned by maximal monomials in lexicographic order. These ideals are central to the
classification in [Mac27] of admissible Hilbert polynomials. Their combinatorial nature
captures geometric information about Hilbert schemes, as shown by [Har66, PS05, Ree95,
RS97] in studying connectedness, radii, and smoothness.
For any vector u := (u ,u ,...,u ) Nn+1, let xu := xu0xu1 xun. The lexicographic
0 1 n ∈ 0 1 ··· n
ordering is the relation > on the monomials in K[x ,x ,...,x ] defined by xu> xv if
lex 0 1 n lex
the first nonzero coordinate of u v Zn+1 is positive, where u,v Nn+1.
− ∈ ∈
Example 3.1. We have x > x > > x in lexicographic order on K[x ,x ,...,x ].
0 lex 1 lex lex n 0 1 n
···
Further, if n 2, then x x2> x4> x3.
≥ 0 2 lex 1 lex 1
For a homogeneous ideal I K[x ,x ,...,x ], lexicographic order gives rise to two
0 1 n
⊂
monomial ideals associated to I. First, the lexicographic ideal for the Hilbert function h
I
in K[x ,x ,...,x ] is the monomial ideal LhI whose i-th graded piece is spanned by the
0 1 n n
h (i) h (i) = dim I largest monomials in K[x ,x ,...,x ] , for all i Z. The
K[x0,x1,...,xn] − I K i 0 1 n i ∈
equality h = h holds by definition, and LhI is a homogeneous ideal of K[x ,x ,...,x ];
I LhnI n 0 1 n
see [Mac27, Section II] or [MS05b, Proposition 2.21]. More importantly, the (saturated)
lexicographic ideal LpI for the Hilbert polynomial p is the monomial ideal
n I
LhI : x ,x ,...,x ∞ := f K[x ,x ,...,x ] f x ,x ,...,x j LhI .
n h 0 1 ni ∈ 0 1 n | h 0 1 ni ⊆ n
j≥1
(cid:0) (cid:1) [(cid:8) (cid:9)
Saturation with respect to the irrelevant ideal x ,x ,...,x K[x ,x ,...,x ] does not
0 1 n 0 1 n
hp i ⊂
affect the Hilbert function in large degrees, so L I also has Hilbert polynomial p .
n I
Example 3.2. If X P2 is three distinct noncollinear points, then the Hilbert function
⊂
of I K[x ,x ,x ] has values h (N) = (1,3,3,3,3,...). The lexicographic ideal in
X 0 1 2 X
⊂
K[x ,x ,x ] for h equals LhX = x2,x x ,x x ,x3 . Therefore, the saturation of LhX with
0 1 2 X 2 h 0 0 1 0 2 1i 2
respect to the irrelevant ideal x ,x ,x is L3 = x ,x3 K[x ,x ,x ], whose Hilbert
h 0 1 2i 2 h 0 1i ⊂ 0 1 2
function has values h (N) = (1,2,3,3,3,3,...).
L3
2
Given a finite sequence of nonnegative integers a ,a ,...,a N, consider the mono-
0 1 n−1
∈
mial ideal L(a ,a ,...,a ) in K[x ,x ,...,x ] with monomial generators
0 1 n−1 0 1 n
xan−1+1,xan−1xan−2+1,...,xan−1xan−2 xa2 xa1+1,xan−1xan−2 xa1 xa0 ;
h 0 0 1 0 1 ··· n−3 n−2 0 1 ··· n−2 n−1i
see [RS97, Notation 1.2]. Lemma 3.3(i) appears in [Moo12, Theorem 2.23].
Lemma 3.3. Let p(t) := d t+i t+i−ei , for integers e e e > 0, and let
i=0 i+1 − i+1 0 ≥ 1 ≥ ··· ≥ d
n N satisfy n > d = degp.
∈ P (cid:0) (cid:1) (cid:0) (cid:1)9
(i) Define e := 0, for d+1 i n, and a := e e , for all 0 j n 1. We have
i j j j+1
≤ ≤ − ≤ ≤ −
Lp = L(a ,a ,...,a )
n 0 1 n−1
= x ,x ,...,x ,xad+1 ,
h 0 1 n−(d+2) n−(d+1)
xad xad−1+1,...,xad xad−1 xa2 xa1+1,xad xad−1 xa1 xa0 .
n−(d+1) n−d n−(d+1) n−d ··· n−3 n−2 n−(d+1) n−d ··· n−2 n−1i
(ii) If there is an integer 0 ℓ d 1 such that a = 0 for all j ℓ, and a > 0, then the
j ℓ+1
≤ ≤ − ≤
minimal monomial generators of Lp are given by m ,m ,...,m , where
n 1 2 n−(ℓ+1)
m := x , for all 1 i n (d+1), and
i i−1
≤ ≤ −
k−1
m := xad−j xad−k+1 , for all 0 k d (ℓ+1).
n−d+k n−(d+1)+j n−(d+1)+k ≤ ≤ −
!
j=0
Y
If a = 0, then the minimal monomial generators are those listed in Part (i).
0
6
Proof.
(i) Substituting the values a := e e determined by the Macaulay–Hartshorne ex-
j j j+1
−
pression of p(t) in the definition of L(a ,a ,...,a ) gives the listed monomials. In
0 1 n−1
degreea +a + +a +1,themonomialxad xad−1 xad−k+1 isthelargest
d d−1 ··· d−k n−(d+1) n−d ··· n−(d+1)+k
monomial smaller than xad xad−1 xad−(k−1)+1 xad−k, for 0 k d 1. In de-
n−(d+1) n−d ··· n−(d+1)+k−1 n ≤ ≤ −
gree a + a + + a , the monomial xad xad−1 xa0 is similarly the largest
d d−1 ··· 0 n−(d+1) n−d ··· n−1
monomial smaller than xad xad−1 xa1+1xa0−1, thus, L := L(a ,a ,...,a ) is
n−(d+1) n−d ··· n−2 n 0 1 n−1
a lexicographic ideal. For any monomial g in the saturation L : x ,x ,...,x ∞ ,
0 1 n
h i
there exists j N such that gxj L. Because the generators of L are not divisible by
∈ n ∈ (cid:0) (cid:1)
x , this implies that g L, showing that L is saturated.
n
∈
Before showing that L has the correct Hilbert polynomial, we first prove that
the auxiliary ideal L′ := L(0,0,...,0,a ,0,0,...,0) = x ,x ,...,x ,xad
d h 0 1 n−(d+2) n−(d+1)i
has Hilbert polynomial d t+i t+i−ad . Setting S := K[x ,x ,...,x ],
i=0 i+1 − i+1 n−(d+1) n−d n
multiplication by xad defines the first homomorphism in a short exact sequence
n−(d+P1) (cid:0) (cid:1) (cid:0) (cid:1)
0 S( a ) S S/ xad 0. Additivity of Hilbert polynomials on short
→ − d → → h n−(d+1)i →
exact sequences shows that p (t) = t+d+1 t+d+1−ad . Applying the summation
L′ d+1 − d+1
formula [GKP94, p. 159] establishes that p (t) = d t+i t+i−ad .
(cid:0) L(cid:1)′ (cid:0) i=0 (cid:1)i+1 − i+1
We prove the general case by induction on d := degp. Suppose that d = 0. The
P (cid:0) (cid:1) (cid:0) (cid:1)
ideal L becomes L := x ,x ,...,x ,xa0 , which has constant Hilbert polynomial
h 0 1 n−2 n−1i
equal to a = p. Suppose that d > 0, let L′ := L(0,0,...,0,a ,0,0,...,0), and let
0 d
L′′ := L(a ,a ,...,a ,0,0,...,0). By the short exact sequence
0 1 d−1
0 (K[x ,x ,...,x ]/L′′)( a ) K[x ,x ,...,x ]/L K[x ,x ,...,x ]/L′ 0,
0 1 n d 0 1 n 0 1 n
→ − → → →
where the injection sends 1 xad , we have p = p +p . Induction yields
7→ n−(d+1) L L′ L′′
p (t) = d−1 t+i−ad t+i−ei , and p (t) = d t+i t+i−ad by the previous
L′′ i=0 i+1 − i+1 L′ i=0 i+1 − i+1
paragraph, so that p = p. Hence, L := L(a ,a ,...,a ) = Lp is the lexicographic
P (cid:0) L(cid:1) (cid:0) (cid:1) 0 1P (cid:0)n−(cid:1)1 (cid:0) n (cid:1)
ideal for p in K[x ,x ,...,x ].
0 1 n
(ii) We know that xad xad−1 xa1 xa0 = xad xad−1 xaℓ+2 xaℓ+1 , because
n−(d+1) n−d ··· n−2 n−1 n−(d+1) n−d ··· n−(ℓ+3) n−(ℓ+2)
either a = a = = a = 0, or a = 0 and ℓ = 1. If ℓ 0, then the monomial
0 1 ℓ 0
··· 6 − ≥
10