The Waldschmidt constant for squarefree monomial ideals

Transcription

1 The Waldschmidt constant for 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

2

3 Outline In this talk we ll focus on a squarefree monomial ideal I with primary decomposition I = s i=1 P i = (x i1,..., x itj ).

4 Outline In this talk we ll focus on a squarefree monomial ideal I with primary decomposition I = s i=1 P i = (x i1,..., x itj ). Algebra symbolic powers initial degree (alpha) Waldschmidt constant

5 Outline In this talk we ll focus on a squarefree monomial ideal I with primary decomposition I = s i=1 P i = (x i1,..., x itj ). Algebra symbolic powers initial degree (alpha) Waldschmidt constant Combinatorics hypergraph (hyper-vertex) coloring fractional chromatic number

6 Outline In this talk we ll focus on a squarefree monomial ideal I with primary decomposition I = s i=1 P i = (x i1,..., x itj ). Algebra symbolic powers initial degree (alpha) Waldschmidt constant Linear Programming Combinatorics hypergraph (hyper-vertex) coloring fractional chromatic number

7 Symbolic powers Definition The n-th symbolic power I (n) of an ideal I R is I (n) = I n R P R P Ass(I ) If I has no embedded primes then I (n) = P n R P R = P Ass(I ) P Ass(I ) P (n) If I has no embedded primes and every P is a complete intersection I (n) = P Ass(I ) P n

8 Growth of the α-invariant Definition For a homogeneous ideal J we denote by α(j) the smallest degree of 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 (m 1+m ) 2 I (m1) I (m2), α(i (m 1+m ) 2 ) α(i (m1) ) + α(i (m2) ) α(i given a subadditive function, lim (m) ) n m = inf α(i (m) ) m exists.

9 Waldschmidt constant Definition Given any homogeneous ideal I, the Waldschmidt constant of I is ˆα(I ) := lim n α(i (n) ). n since α(i (n) ) nα(i ), we have ˆα(I ) α(i ) by Ein-Lazarsfeld-Smith, Hochster-Huneke, if e =big-height(i ) I (em) I m, α(i (em) ) mα(i ) ˆα(I ) α(i ) e

10 Computing ˆα for a squarefree An Example monomial ideal Example: I = (xy, xz, yz) = (x, y) (x, z) (y, z) I =(x 0 x 1, x 0 x 2, x 1 x 2 )=(x 0, x 1 ) \ (x 0, x 2 ) \ (x 1, x 2 ) k[ Q = conv(x 0, x 1 ) \ conv(x 0, x 2 ) \ conv(x 1, x 2 ). conv(x 0, x 1 ) conv(x 0, x 2 ) conv(x 1, x 2 ) x 0 0 x 0 0 x 1 0 x 1 0 x 2 0 x 2 0 x 0 + x 1 1 x 0 + x 2 1 x 1 + x 2 1 x a y b z c I (n) = (x, y) n (x, z) n (y, z) n a a + b n n a + c n + b n 1 a n + c n 1 b b + c n n a, b, c 0 + c n 1 a n, b n, c n 0 Figure: Q(I ) { a ˆα(I ) = min n + b n + c ( a n n, b n, c ) } Q(I ) = 3 n 2

11 A linear programming approach Lemma (BCGHJNSVV) Let I = P 1 P 2... P s be a squarefree monomial ideal and { 1 if x A i,j = j P i 0 if x j P i. Then ˆα(I ) is the optimum value of the LP minimize 1 T y subject to Ay 1 and y 0. In particular, for a monomial ideal, ˆα(I ) Q.

12 Waldschmidt constant computed... in two ways 1 as a limit α(i ˆα(I ) = lim (n) ) α(i n n = inf (n) ) n n 2 as the optimum value of a linear program minimize 1 T y subject to Ay 1 and y 0. These two quantities are equal by our theorem.

13 Enter hypergraphs Definition There is a 1-to-1 corespondence between hypergraphs H = (V, E) and squarefree monomial ideals I (H) given by {x i1,..., x it } E x i1 x it is a minimal generator of I (H). Example: I (H) = (xy, xz, yz)

14 Fractional chromatic number Preface 3:00 Figure B: A schedule for the five committees. ix 12: : Scheduling 5 committees 1:00 2:00 1:00 2:00 3:00 3:00 Figure B: A schedule for the Figure five committees. C: An improved schedule for the five committees. Coloring C 5 Chromatic number on this subject. 1 2 In3 the 4 course 5 of writing this book we found that Claude Berge 12:00 monograph [16] on this very subject. Berge s Fractional Graph Theory is based delivered at the Indian Statistical Institute twenty years ago. Berge includes a tr fractional 1:00matching number and the fractional edge chromatic number. 1 Two dec agreatdealofdevelopmentinthefieldoffractionalgraphtheoryandthetimeis overview. 2:00 χ(c 5 ) = 3 χ f (C 5 ) = 5 2 = 2.5 Rationalization 3:00 Figure C: AnWe improved have two schedule principal formethods the five committees. to convert graph concepts from integer to fractio is to formulate the concepts as integer programs and then to consider the linear relaxation (see A.3). The second is to make use of the subadditivity lemma (Le

15 Fractional chromatic number defined... in two ways 1 If H is a hypergraph with maximal independent sets {W 1,..., W t }, the fractional chromatic number χ f (H) is the optimum value for minimize 1 T y subject to By 1 and y 0. { 1 if x where B i,j = i W j 0 if x i W j. 2 χ f (H) = lim b χ b (G) b = inf b χ b (G) b. These two quantities are equal by general machinery.

16 Waldschmidt fractional chromatic duality Waldschmidt constant If I = P 1 P 2... P s, then ˆα(I ) = the optimum value for minimize 1 T y subject to Ay 1 y 0. where A i,j = { 1 if x j P i 0 if x j P i. Fractional chromatic # If H is a hypergraph with maximal independent sets {W 1,..., W t }, χ f (H) = the optimum value for minimize 1 T y subject to By 1 y 0. where B i,j = { 1 if x i W j 0 if x i W j. Theorem (Bocci,Cooper,Guardo,Harbourne,Janssen,Nagel, S.,VanTuyl,Vu) α(i ) = χ f (H(I )) χ f (H(I )) 1.

17 Consequences Corollary (BCGHJNSVV) Let G be a graph with chromatic number χ(g) and clique number ω(g) (thus ω(g) χ f (G) χ(g)). (i) Then χ(g) α(i (G)) ω(g) χ(g) 1 (ii) If G is a perfect graph, then α(i (G)) = ω(g) 1. χ(g) χ(g) 1. (iii) If G is a complete k-partite graph, then α(i (G)) = (iv) If G is bipartite, then α(i (G)) = 2. k k 1. (v) If G = C 2n+1 is an odd cycle, then α(i (C 2n+1 )) = 2n+1 n+1. (vi) If G = C c 2n+1, then α(i (G)) = 2n+1 2n 1.