TY - JOUR

T1 - On the Dual Graphs of Cohen-Macaulay Algebras

AU - Benedetti, Bruno

AU - Varbaro, Matteo

N1 - Publisher Copyright:
© 2014 The Author(s).

PY - 2015

Y1 - 2015

N2 - Given an equidimensional algebraic set X⊆Pn, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: (1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible. For coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) (2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in Pn, then the diameter of the graph G(X) is less than or equal to codimPnX. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.) On the way to these results, we show that there exists a graph which is not the dual graph of any simplicial complex (no matter the dimension).

AB - Given an equidimensional algebraic set X⊆Pn, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshorne's connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshorne's result: (1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible. For coordinate arrangements, it yields an algebraic extension of Balinski's theorem for simplicial polytopes.) (2) If X is an arrangement of lines no three of which meet in the same point, and X is canonically embedded in Pn, then the diameter of the graph G(X) is less than or equal to codimPnX. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.) On the way to these results, we show that there exists a graph which is not the dual graph of any simplicial complex (no matter the dimension).

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U2 - 10.1093/imrn/rnu191

DO - 10.1093/imrn/rnu191

M3 - Article

AN - SCOPUS:84943562828

VL - 2015

SP - 8085

EP - 8115

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 17

ER -