Susan M. Cooper - Research Statement
1. Introduction and Overview
My research is motivated by the links between algebraic geometry and commutative algebra1. My work
has involved topics such as determinantal schemes [10] and enumerating semidualizing modules [14]. I
am especially interested in obtaining geometric information about a scheme in projective n-space2, Pn ,
using tools which encode algebraic invariants associated to the homogeneous ideal defining the scheme.
Examples of applications come from areas such as coding theory, graph theory, and statistics. These connections lead to exciting research and collaborations for all levels from the undergraduate to the expert.
To introduce two key invariants, consider the polynomial ring R = k[ x0 , . . . , xn ] where k is an algebraically closed field of characteristic 0. We say that F ∈ R is homogeneous if every term of F has the
same degree. An ideal I ⊆ R is homogeneous if it can be generated by a set of homogeneous polynomials.
We group the elements of I by degree which results in a collection of finite dimensional vector spaces. For
example, let I = ( F, G ) ⊆ C[ x0 , x1 , x2 ] where F = x0 − x2 and G = x12 − x1 x2 . The set { x0 F, x1 F, x2 F, G }
forms a C-basis for the space of degree 2 homogeneous polynomials in I, and so we write dimC I2 = 4.
The degree-by-degree dimensions dimk Id of a homogeneous ideal I ⊆ R give a function of d, called the
Hilbert function, introduced by David Hilbert [30] in his work in invariant theory.
Definition 1.1. Let I ⊆ R be a homogeneous ideal. The Hilbert function of R/I, denoted H ( R/I ), is the
function where H ( R/I, d) := dimk ( R/I )d = dimk Rd − dimk Id = (n+d d) − dimk Id for d ≥ 0.
Related to the Hilbert function of a homogeneous ideal I ⊆ R are invariants obtained by looking at the
relations on the generators of I, and the relations on these relations (called the syzygies), etc. When we
study these relations we find an exact sequence
0→
M
i
R(−i ) β l,i ( I ) →
M
i
R(−i ) β l −1,i ( I ) → · · · →
M
R(−i ) β0,i ( I ) → I → 0,
i
where R(−i ) is the ring R but with a shift in the grading and the maps are given by matrices of homogeneous polynomials of positive degree. The numbers β j,i ( I ) are the graded Betti numbers for I.
Question 1.2. What does the zero-locus of an ideal tell us about its Hilbert function and Betti numbers?
My research has been driven by this type of question with a focus on ideals of point sets. On a first encounter, 0-dimensional schemes may seem to be devoid of substance. However, to the contrary, the study
of points has intrigued algebraic geometers for hundreds of years. For example, Castelnuovo showed that
non-trivial information about a curve can be retrieved from the points arising as the general hyperplane
sections of the curve [19]. Although the study of points has a deep history, it manages to host many exciting open problems [5]. This note focuses on my past, current, and future work in the following settings:
(a) The Hilbert functions of fat point subschemes (defined below) are not well-understood. However, recently in joint work with B. Harbourne and Z. Teitler [11] we produce easy to compute
upper and lower bounds for Hilbert functions of fat points in P2 and give an explicit criterion for
when these bounds coincide. The result is then applied to obtain upper and lower bounds for
graded Betti numbers. Although still largely unexplored, the underlying approach can be applied
to give bounds for fat points in any projective space. These results are proving to be fruitful in considering a conjecture arising from work of B. Harbourne and C. Huneke about the initial degree of
a fat point scheme. This approach is also being applied to fat points in multi-projective spaces.
(b) The Eisenbud-Green-Harris Conjecture concerns characterizing Hilbert functions of homogeneous
ideals containing regular sequences. In my Ph.D. dissertation I proved the conjecture when restricted to certain ideals of finite sets of distinct points. In doing so I applied the Cayley-Bacharach
Theorem which relates the Hilbert functions of three sets of points. I am investigating the possibility of a similar formula for fat points. I am also studying geometric consequences of Hilbert
functions exhibiting extremal behavior related to natural bounds arising in these situations.
1
MSC2010: 13C05, 13C14, 13C40, 13D02, 13D40, 13H15, 13P99, 14C99, 14M06, 14M12.
2Pn is the set of non-zero (n + 1)-tuples of elements of a field k modulo the relation (s , . . . , s ) ∼ (ts , . . . , ts ) for 0 6= t ∈ k.
n
n
0
0
Susan M. Cooper - Research Statement
2. Underlying Motivation
Hilbert functions have been extensively studied and have played a central role in commutative algebra
and algebraic geometry. Perhaps the most celebrated result in the algebraic setting is Macaulay’s Theorem.
Definition 2.1. A lex ideal is a monomial ideal L ⊆ R = k[ x0 , . . . , xn ] which is minimally generated by
monomials {m1 , . . . , mr }, where, for j = 1, . . . , r, all monomials of degree deg(m j ) which are larger than
m j in the degree-lexicographic ordering are contained in L.
Given two natural numbers a and d we can define a unique natural number, denoted a<d> , in terms
of the d-binomial expansion of a (for details, see pages 159 - 161 of [3]). The function <d> turns out to
describe the growth of lex ideals from one degree to the next. Macaulay’s Theorem says that this behavior
holds for all homogeneous ideals.
Theorem 2.2. [3, 33, 39] (Macaulay’s Theorem) Let I ⊆ R = k[ x0 , . . . , xn ] be a homogeneous ideal.
(a) There exists a lex ideal L ⊆ R such that H ( R/I ) = H ( R/L).
(b) For any lex ideal L ⊆ R we have the bounds H ( R/L, d + 1) ≤ H ( R/L, d)<d> for d ≥ 0. Thus,
H ( R/I, d + 1) ≤ H ( R/I, d)<d> for d ≥ 0.
To demonstrate, let I = ( x02 + x0 x1 , x03 x1 , x05 + x15 ) ⊆ R = k[ x0 , x1 ]. Then dimk I1 = 0, dimk I2 =
1, dimk I3 = 2, dimk I4 = 4, and dimk Ii = dimk Ri for i ≥ 5. Let L = L1 + L2 + · · · be the lex ideal where
Li is generated by the largest dimk Ii monomials in the degree-lexicographic ordering; so L1 = (0), L2 =
( x02 ), L3 = ( x03 , x02 x1 ), L4 = ( x04 , x03 x1 , x02 x12 , x0 x13 ), etc. Then H ( R/L) = H ( R/I ) = (1, 2, 2, 2, 1, 0, . . .). Moreover, by part (b) of Macaulay’s Theorem, H ( R/I, d + 1) ≤ H ( R/I, d)<d> for d ≥ 0.
Many people have tried to characterize the possible Hilbert functions of families of homogeneous ideals
I whose elements share special properties. Approaches have included determining the ideals L which play
analogous roles to the lex ideals in part (a) of Macaulay’s Theorem and describing the Hilbert functions
combinatorially by giving bounds on each H ( R/I, d) as in part (b) of Macaulay’s Theorem. Indeed, these
characterizations have developed in a variety of different directions (for some examples see [1, 22, 31, 32,
37]). On the geometric side, we can consider Hilbert functions of points.
Definition 2.3. If X ⊆ Pn is a set of points and I(X) ⊆ R = k[ x0 , . . . , xn ] is the ideal consisting of all the
homogeneous polynomials vanishing on X, then the Hilbert function of X is H (X) := H ( R/I(X)).
Geramita-Maroscia-Roberts [24] characterize the functions which arise as Hilbert functions of finite sets
of distinct, reduced points in Pn . Empowered with such a characterization, the Hilbert function of a set
of points X can then be exploited to obtain both algebraic data about I(X) and geometric information
about X. For example, E. D. Davis [15] and Bigatti-Geramita-Migliore [2] use extremal behavior of Hilbert
functions (described by part (b) of Theorem 2.2) to guarantee that some subsets have special properties.
3. Hilbert Functions, Graded Betti Numbers, and Initial Degrees of Fat Points
The determination of Hilbert functions and graded Betti numbers for ideals of fat point schemes are
issues of significant interest in algebraic geometry.
Definition 3.1. Let X = { P1 , . . . , Pr } be a finite set of distinct points in Pn and m1 , . . . , mr be positive
integers. The fat point subscheme Y = m1 P1 + · · · + mr Pr of Pn is defined by the homogeneous ideal
I = I( P1 )m1 ∩ I( P2 )m2 ∩ · · · ∩ I( Pr )mr ⊆ R = k[ x0 , . . . , xn ].
We call X = { P1 , . . . , Pr } the support of Y and m1 , . . . , mr the multiplicities of Y.
Question 3.2. Given positive integers m1 , . . . , mr , is there a characterization of the functions that occur as
Hilbert functions of fat points in Pn whose multiplicities are m1 , . . . , mr ? What are all the possible graded
Betti numbers?
Whereas for points in Pn with n = 1 the Hilbert function depends only on the multiplicities, for n > 1
the Hilbert function depends also on the position of the points, and in a very subtle way. Indeed, Question 3.2 is open in P2 with each mi = 2 (called double point schemes). However, if r ≤ 8 then we know
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Susan M. Cooper - Research Statement
all the possible Hilbert functions for any fat point scheme in P2 for any given set of multiplicities [23, 27]. In
addition, one of the main results of Geramita-Migliore-Sabourin in [25] is to give a criterion characterizing
a family of functions, all of which occur as Hilbert functions of double point schemes in P2 .
In [11], B. Harbourne, Z. Teitler and I study the Hilbert function of a fat point scheme Y ⊂ P2 given
the multiplicities and data about which subsets of the support are collinear. We define a reduction vector
d based on collinearity data and use d to give upper and lower bounds on H (Y) in each degree. Here is a
brief description of our approach.
Definition 3.3. [11] Let Y = m1 P1 + m2 P2 + · · · + mr Pr ⊂ P2 .
(1) Let L be a line in P2 . We define deg(L ∩ Y) := ∑ Pi ∈L mi . The subscheme of Y residual to L is
Y0 := b1 P1 + b2 P2 + · · · + br Pr , where bi = mi if Pi 6∈ L and bi = max(mi − 1, 0) if Pi ∈ L.
(2) We say that a sequence L1 , . . . , Ln+1 of lines totally reduces Y if at least m j of the lines Li contain
Pj for each j. To such a sequence of lines we associate the reduction vector d = (d1 , . . . , dn+1 ) with
di = deg(Li ∩ Yi−1 ), where Y = Y0 and Yi is the residual of Yi−1 with respect to Li .
The lower bound given in [11] on the Hilbert function of a fat point subscheme in P2 can be described
combinatorially via a simple diagram.
Definition 3.4. Let d = (d1 , . . . , dn+1 ) be a non-negative integer vector.
(1) The standard configuration Sd determined by d consists of the d j leftmost first quadrant lattice
points of Z × Z on each horizontal line with second coordinate j − 1 for 1 ≤ j ≤ n + 1.
(2) We define the function f d where f d (t) for t ≥ 0 is the number of points in Sd which are in or on the
triangle whose vertices are (0, 0), (t, 0) and (0, t).
Example 3.5. Let d = (8, 6, 5, 2). Then Sd is the set of lattice points in Z × Z shown in the figure below.
y
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
u
x
In this example, f d = (1, 3, 6, 10, 14, 17, 20, 21, 21, 21, . . .).
In a similar, but somewhat more complicated way, one can define a function Fd which has the property
that Fd ≥ f d . We also have a simple criterion to determine exactly when f d = Fd . In particular, if d =
(d1 , . . . , dn+1 ) such that di > di+1 then f d = Fd .
Theorem 3.6. [11] If L1 , . . . , Ln+1 is any sequence of lines which totally reduces a fat point subscheme
Y ⊂ P2 with reduction vector d, then
f d (t) ≤ H (Y, t) ≤ Fd (t)
for all t ≥ 0. Moreover, for example, if the entries of d are strictly decreasing, then H (Y) = f d .
Using similar methods we also give upper and lower bounds on the graded Betti numbers for I(Y).
For future related projects, we are naturally led to the following questions.
Questions for Current & Future Research. Suppose Y = m1 P1 + · · · + mr Pr is a fat point scheme in P2
with an associated reduction vector d for some totally reducing sequence of lines.
• Different choices of lines will produce different upper and lower bounds f d and Fd . How do the
bounds change with respect to these choices?
• How sharp are the bounds f d and Fd ?
• What can be said about H (Y) if we apply residuation using higher degree curves?
• If f d 6= Fd , can we nonetheless find a formula for H (Y)?
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Susan M. Cooper - Research Statement
We conclude this section with an application to initial degree. Related to the Hilbert function and graded
Betti numbers, there are several numerical characters that have gained the attention of many researchers
over the years. The paper [28] provides a nice list of important numerical characters and progress on
problems related to them. One of the more important such characters is α.
Definition 3.7. Let Y = m1 P1 + · · · + mr Pr be a fat point subscheme of Pn . Then we define the initial
degree of Y, denoted α(Y), to be the least degree t such that I(Y)t 6= 0.
In unpublished work, B. Harbourne and C. Huneke [29] compare symbolic and regular powers of
homogeneous ideals in a more general setting. This has led them to ask if the following conjecture is true:
Conjecture 3.8. Let R = k[P2 ] = k[ x0 , x1 , x2 ] and consider X = P1 + · · · + Pr and Y = ∑ri=1 (2s − 1) Pi .
Then I(Y) ⊂ ( x0 , x1 , x2 )s−1 I(X)s .
Conjecture 3.8 would imply the following conjecture.
Conjecture 3.9. Let X = P1 + · · · + Pr ⊂ P2 and Y = ∑ri=1 (2s − 1) Pi ⊂ P2 . Then α(Y) ≥ sα(X) + (s − 1).
Conjecture 3.9 is known to be true for special sets of points (such as when X is a complete intersection).
As further evidence, S. Hartke and I [12] have verified Conjecture 3.9 for a large collection of point sets
within the family of line count configurations. In doing so we invoke the results of [11] and work with
the upper and lower bounds associated with H (X) and H (Y).
4. The Eisenbud-Green-Harris Conjecture for Points
The Eisenbud-Green-Harris Conjecture is an attempt to generalize Macaulay’s Theorem for ideals containing regular sequences. Technically speaking, { F1 , . . . , Fr } in R = k[ x0 , . . . , xn ] is a regular sequence of
forms if each Fi is a homogeneous polynomial such that Fi+1 is not a zero-divisor on R/( F1 , . . . , Fi ) R and
( F1 , . . . , Fr ) 6= R. For example, if 2 ≤ a0 ≤ a1 ≤ · · · ≤ an are integers then { x0a0 , x2a2 , . . . , xnan } is a regular
sequence. These monomials play a key role in the analog of the lex ideal from Macaulay’s Theorem.
Definition 4.1. Let A := { a0 , a1 , . . . , an } where 2 ≤ a0 ≤ a1 ≤ · · · ≤ an are integers. A lex-plus-powers
ideal with respect to A is a monomial ideal in R of the form J + ( x0a0 , x1a1 , . . . , xnan ) where J is a lex ideal.
For example, let A = {2, 3, 3}. Then L1 = ( x02 , x13 , x23 , x0 x12 , x0 x1 x2 ) is a lex-plus-powers ideal with respect
to A, but L2 = ( x02 , x13 , x23 , x0 x12 , x0 x1 x2 , x12 x2 ) is not since x0 x22 >d−lex x12 x2 and x0 x22 6∈ L2 . Motivated by the
combinatorial work of Clements-Lindström [6], L. Roberts and I have shown:
Theorem 4.2. [7, 13] If I ⊆ R is any homogeneous ideal containing { x0a0 , . . . , xnan }, then there is a lex-pluspowers ideal L ⊆ R with respect to A = { a0 , . . . , an } such that H ( R/I ) = H ( R/L).
Eisenbud-Green-Harris were interested in homogeneous ideals containing a regular sequence that is
not necessarily equal to the special sequence { x0a0 , . . . , xnan }. They conjectured the following:
Conjecture 4.3. [4, 17, 18, 21] (EGH Conjecture) If I ⊆ R is a homogeneous ideal containing a regular
sequence F0 , . . . , Fn of forms of degrees deg( Fi ) = ai , then there is a lex-plus-powers ideal L ⊆ R with
respect to A = { a0 , . . . , an } such that H ( R/I ) = H ( R/L).
I used the work of Greene-Kleitman [26] to prove:
Theorem 4.4. [8] The EGH Conjecture is true when I ⊆ R = k[ x0 , x1 , x2 ] has minimal generators which
are all in the same degree such that two of the minimal generators form a regular sequence in k[ x0 , x1 ].
Despite much effort, the EGH Conjecture is known to be true only in some exceptional cases. The
conjecture has been proven in the cases where L is an almost complete intersection [20], and when n = 1
[7, 38]. Mermin-Peeva-Stillman [35] have results for ideals containing squares of the variables. In addition,
j −1
Caviglia & Maclagan [4] have proven that the EGH Conjecture is true if a j > ∑i=0 ( ai − 1) for j ≥ 1. Joint
work of Mermin & Murai [34] has shown the conjecture holds for pure powers. On the geometric side, I
[7] have studied the conjecture for ideals of points (see Theorem 4.7).
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Susan M. Cooper - Research Statement
Definition 4.5. A complete intersection in Pn is a set of points Y ⊆ Pn such that I(Y) is generated by
n homogeneous polynomials which form a regular sequence. The list {d1 , . . . , dn } of the degrees of the
generating homogeneous polynomials is the type of the complete intersection.
A finite set of distinct, reduced points is contained in some (in fact, many) complete intersections, if
we have the freedom to pick the degrees of the defining polynomials. A main idea that has motivated
my research is to fix these degrees and find conditions that a given Hilbert function must satisfy in order
for it to be the Hilbert function of a subset of points in a complete intersection. Motivated by the EGH
Conjecture I focused my attention on complete intersections of the form
{[1 : a1 : . . . : an ] | a1 , . . . , an ∈ Z, 0 ≤ a1 ≤ dn − 1, 0 ≤ a2 ≤ dn−1 − 1, . . . , 0 ≤ an ≤ d1 − 1} ⊆ Pn .
This set is called a rectangular complete intersection of type {d1 , . . . , dn }, denoted RectCI(d1 , . . . , dn ).
Bounds similar to those in part (b) of Macaulay’s Theorem and resulting from the work of GreeneKleitman [26] can be applied to characterize the Hilbert functions of subsets of RectCI(d1 , . . . , dn ).
Question 4.6. Is it true that { H (X) | X ⊆ Y ∈ CI (d1 , . . . , dn )} = { H (W) | W ⊆ RectCI (d1 , . . . , dn )}?
I have shown that Question 4.6 is equivalent to the EGH Conjecture restricted to ideals of points.
Theorem 4.7. [7, 9] The answer to Question 4.6 is “yes” in the cases:
(1) n = 2
(2) n = 3 in the situations: (a) d1 = 2; (b) d1 = 3; (c) 4 ≤ d1 ≤ d2 ≤ d3 and d3 ≥ d1 + d2 − 1.
There is still much work to be done on the EGH Conjecture. One question that I find intriguing is:
Question for Current & Future Research. Essential parts of the proofs for Theorem 4.7 rely on algebraic
and geometric consequences of when Hilbert functions achieve the combinatorial bounds resulting from
part (b) of Macaulay’s Theorem. Restricting to subsets of complete intersections, what are the consequences for Hilbert functions exhibiting extremal behavior as described by Greene-Kleitman [26]?
We close this section with questions that not only guide my current and future research program but
connect my work on the EGH Conjecture and fat points.
Question for Current & Future Research. Let X ⊆ Y = RectCI(d1 , d2 ). What are all the Hilbert functions
that arise from adding multiplicities to the points of X?
Since we know that every Hilbert function of a reduced subset of any complete intersection in P2 is
achieved as the Hilbert function of some reduced subset of a rectangular complete intersection in P2 , it is
natural to try to classify associated fat points in a similar fashion. More precisely, we ask:
Question for Current & Future Research. Suppose X is a reduced subset of some complete intersection
of type {d1 , d2 }. Attach multiplicities to the points in X to obtain a fat point scheme W. Is there a fat point
scheme W0 whose support is a subset X0 ⊆ RectCI(d1 , d2 ) such that H (W) = H (W0 )?
Finally, one machine that is useful in working with subsets of complete intersections comes from liaison.
Theorem 4.8. [4, 16, 36] (Cayley-Bacharach Theorem) Let J = ( F1 , . . . , Fn ) ⊆ S = k[ x1 , . . . , xn ] be a homogeneous ideal where { F1 , . . . , Fn } is a regular sequence with deg( Fi ) = di . Let I be a homogeneous ideal
containing J and let s = ∑in=1 (di − 1). Then for 0 ≤ t ≤ s we have
H (S/J, t) = H (S/I, t) + H (S/( J : I ), s − t).
As a consequence, if X ⊆ Y ∈ CI (d1 , . . . , dn ) then Theorem 4.8 relates H (X), H (Y) and H (Y − X).
Question for Current & Future Research. Is there an analog to the Cayley-Bacharach Theorem for fat
points?
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Susan M. Cooper - Research Statement
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