The Waldschmidt constant for squarefree monomial idealsaseceleanu2/research/Fractional.pdf · 2016....

The Waldschmidt constantfor

squarefree monomial ideals

Alexandra Seceleanu(joint with C. Bocci, S. Cooper, E. Guardo, B. Harbourne, M. Janssen, U. Nagel, A. Van Tuyl, T. Vu)

University of Nebraska-Lincoln

Alexandra Seceleanu (UNL) The Waldschmidt constant for squarefree monomial ideals

Outline

In this talk we’ll focus on a squarefree monomial ideal I withprimary decomposition I =

⋂si=1 Pi =

⋂(xi1 , . . . , xitj

).


Outline


⋂si=1 Pi =

⋂(xi1 , . . . , xitj

).

Algebra

symbolic powers

initial degree(alpha)

Waldschmidtconstant


Outline


⋂si=1 Pi =

⋂(xi1 , . . . , xitj

).

Algebra

symbolic powers


Waldschmidtconstant

Combinatorics

hypergraph

(hyper-vertex)coloring

fractionalchromatic number


Outline


⋂si=1 Pi =

⋂(xi1 , . . . , xitj

).

Algebra

symbolic powers


Waldschmidtconstant

LinearProgramming

Combinatorics

hypergraph

(hyper-vertex)coloring

fractionalchromatic number


Symbolic powers

Definition

The n-th symbolic power I (n) of an ideal I ⊂ R is

I (n) =⋂

P∈Ass(I )

I nRP ∩ R

If I has no embedded primes then

I (n) =⋂

P∈Ass(I )

PnRP ∩ R =⋂

P∈Ass(I )

P(n)

If I has no embedded primes and every P is a complete intersection

I (n) =⋂

P∈Ass(I )

Pn


Growth of the α-invariant

Definition

For a homogeneous ideal J we denote by α(J) the smallest degreeof an element in a minimal set of homogeneous generators for J.

Measuring the growth for symbolic powers:

α(I (m)) measures the growth of the degrees of elements in I (n)

α(I (m)) is a sub-additive function: since I (m1+m2) ⊇ I (m1)I (m2),

α(I (m1+m2)) ≤ α(I (m1)) + α(I (m2))

given a subadditive function, limn→∞α(I

(m))

m = inf α(I(m)

)m exists.


Waldschmidt constant

Definition

Given any homogeneous ideal I , the Waldschmidt constant of I is

α̂(I ) := limn→∞

α(I (n))

n.

since α(I (n)) ≤ nα(I ), we have α̂(I ) ≤ α(I )

by Ein-Lazarsfeld-Smith, Hochster-Huneke, if e =big-height(I )

I (em) ⊆ Im,

α(I (em)) ≥ mα(I )

α̂(I ) ≥ α(I )

e


Computing α̂ for a squarefree monomial ideal

Example: I = (xy , xz , yz) = (x , y) ∩ (x , z) ∩ (y , z)

xaybzc ∈ I (n) = (x , y)n ∩ (x , z)n ∩ (y , z)n

⇔

a + b ≥ n

a + c ≥ n

b + c ≥ n

a, b, c ≥ 0

⇔

an + b

n ≥ 1an + c

n ≥ 1bn + c

n ≥ 1an ,

bn ,

cn ≥ 0

Alpha and the Waldschmidt Constant

A Polyhedral Approach

An Example

I = (x0x1, x0x2, x1x2) = (x0, x1) \ (x0, x2) \ (x1, x2) ⇢ k[x0, x1, x2]

Q = conv(x0, x1) \ conv(x0, x2) \ conv(x1, x2).

conv(x0, x1) conv(x0, x2) conv(x1, x2)

x0 � 0 x0 � 0 x1 � 0x1 � 0 x2 � 0 x2 � 0

x0 + x1 � 1 x0 + x2 � 1 x1 + x2 � 1

Figure: Q(I )

α̂(I ) = min

{a

n+

b

n+

c

n|(a

n,b

n,c

n

)∈ Q(I )

}=

3

2


A linear programming approach

Lemma (BCGHJNSVV)

Let I = P1 ∩ P2 ∩ . . . ∩ Ps be a squarefree monomial ideal and

Ai,j =

{1 if xj ∈ Pi

0 if xj 6∈ Pi .

Then α̂(I ) is the optimum value of the LP

minimize 1Tysubject to Ay ≥ 1 and y ≥ 0.

In particular, for a monomial ideal, α̂(I ) ∈ Q.


Waldschmidt constant computed

. . . in two ways

1 as a limit

α̂(I ) = limn→∞α(I

(n))

n = infn→∞α(I

(n))

n

2 as the optimum value of a linear program

minimize 1Tysubject to Ay ≥ 1 and y ≥ 0.

These two quantities are equal by our theorem.


Enter hypergraphs

Definition

There is a 1-to-1 corespondence between hypergraphs H = (V ,E )and squarefree monomial ideals I (H) given by

{xi1 , . . . , xit} ∈ E ⇐⇒ xi1 · · · xit is a minimal generator of I (H).

Example: I (H) = (xy , xz , yz)


Fractional chromatic number

Scheduling 5committees

Preface ix

1 2 3 4 512:00

1:00

2:00

3:00

Figure B: A schedule for the five committees.

1 2 3 4 512:00

1:00

2:00

3:00

Figure C: An improved schedule for the five committees.

on this subject. In the course of writing this book we found that Claude Berge wrote a shortmonograph [16] on this very subject. Berge’s Fractional Graph Theory is based on his lecturesdelivered at the Indian Statistical Institute twenty years ago. Berge includes a treatment of thefractional matching number and the fractional edge chromatic number.1 Two decades have seena great deal of development in the field of fractional graph theory and the time is ripe for a newoverview.

Rationalization

We have two principal methods to convert graph concepts from integer to fractional. The firstis to formulate the concepts as integer programs and then to consider the linear programmingrelaxation (see §A.3). The second is to make use of the subadditivity lemma (Lemma A.4.1 onpage 137). It is very pleasing that these two approaches typically yield the same results. Nearlyevery integer-valued invariant encountered in a first course in graph theory gives rise to a fractionalanalogue.

Most of the fractional definitions in this book can be obtained from their integer-valued coun-terparts by replacing the notion of a set with the more generous notion of a “fuzzy” set [190]. Whilemembership in a set is governed by a {0, 1}-valued indicator function, membership in a fuzzy set isgoverned by a [0, 1]-valued indicator function. It is possible to devise fractional analogues to nearly

1More recently, Berge devotes a chapter of his monograph Hypergraphs: Combinatorics of Finite Sets [19] tofractional transversals of hypergraphs, which includes an exploration of fractional matchings of graphs.

Preface ix

1 2 3 4 512:00

1:00

2:00

3:00

Figure B: A schedule for the five committees.

1 2 3 4 512:00

1:00

2:00

3:00

Figure C: An improved schedule for the five committees.

on this subject. In the course of writing this book we found that Claude Berge wrote a shortmonograph [16] on this very subject. Berge’s Fractional Graph Theory is based on his lecturesdelivered at the Indian Statistical Institute twenty years ago. Berge includes a treatment of thefractional matching number and the fractional edge chromatic number.1 Two decades have seena great deal of development in the field of fractional graph theory and the time is ripe for a newoverview.

Rationalization

We have two principal methods to convert graph concepts from integer to fractional. The firstis to formulate the concepts as integer programs and then to consider the linear programmingrelaxation (see §A.3). The second is to make use of the subadditivity lemma (Lemma A.4.1 onpage 137). It is very pleasing that these two approaches typically yield the same results. Nearlyevery integer-valued invariant encountered in a first course in graph theory gives rise to a fractionalanalogue.

Most of the fractional definitions in this book can be obtained from their integer-valued coun-terparts by replacing the notion of a set with the more generous notion of a “fuzzy” set [190]. Whilemembership in a set is governed by a {0, 1}-valued indicator function, membership in a fuzzy set isgoverned by a [0, 1]-valued indicator function. It is possible to devise fractional analogues to nearly

1More recently, Berge devotes a chapter of his monograph Hypergraphs: Combinatorics of Finite Sets [19] tofractional transversals of hypergraphs, which includes an exploration of fractional matchings of graphs.

Coloring C5

Chromaticnumber

χ(C5) = 3 χf (C5) = 52 = 2.5


Fractional chromatic number defined

. . . in two ways

1 If H is a hypergraph with maximal independent sets{W1, . . . ,Wt}, the fractional chromatic number χf (H) is theoptimum value for

minimize 1Tysubject to By ≥ 1 and y ≥ 0.

where Bi ,j =

{1 if xi ∈Wj

0 if xi 6∈Wj .

2 χf (H) = limb→∞χb(G)

b = infbχb(G)

b .

These two quantities are equal by general machinery.


Waldschmidt – fractional chromatic duality

Waldschmidt constantIf I = P1 ∩ P2 ∩ . . . ∩ Ps , thenα̂(I ) = the optimum value for

minimize 1Tysubject to Ay ≥ 1

y ≥ 0.

where Ai,j =

{1 if xj ∈ Pi

0 if xj 6∈ Pi .

Fractional chromatic #If H is a hypergraph with maximalindependent sets {W1, . . . ,Wt},χf (H) = the optimum value for

minimize 1Tysubject to By ≥ 1

y ≥ 0.

where Bi,j =

{1 if xi ∈Wj

0 if xi 6∈Wj .

Theorem (Bocci,Cooper,Guardo,Harbourne,Janssen,Nagel, S. ,VanTuyl,Vu)

α̂(I ) =χf (H(I ))

χf (H(I ))− 1.


Consequences

Corollary (BCGHJNSVV)

Let G be a graph with chromatic number χ(G ) and clique numberω(G ) (thus ω(G ) ≤ χf (G ) ≤ χ(G )).

(i) Thenχ(G )

χ(G )− 1≤ α̂(I (G )) ≤ ω(G )

ω(G )− 1.

(ii) If G is a perfect graph, then α̂(I (G )) = χ(G)χ(G)−1 .

(iii) If G is a complete k-partite graph, then α̂(I (G )) = kk−1 .

(iv) If G is bipartite, then α̂(I (G )) = 2.

(v) If G = C2n+1 is an odd cycle, then α̂(I (C2n+1)) = 2n+1n+1 .

(vi) If G = C c2n+1, then α̂(I (G )) = 2n+1

2n−1 .


The Waldschmidt constant for squarefree monomial idealsaseceleanu2/research/Fractional.pdf · 2016....

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