TY - JOUR
T1 - On the Dual Graphs of Cohen-Macaulay Algebras
AU - Benedetti, Bruno
AU - Varbaro, Matteo
N1 - Publisher Copyright:
© 2014 The Author(s).
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.
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 -