Annals of Combinatorics 1 (1997) 197-213 Annals of Combinatorics Springer-Verlag 1997 Borel Sets and Sectional Matrices* A.M. Bigatti and L. Robbiano Dipartimento di Matematica, Via Dodecaneso 35,1-16146 Genova, Italy {bigatti; robbiano} @dima.unige.it Received July 12, 1997 AMS Subject Classification: 13D40, 05A 15, 14N 10 Abstract. Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial struc- ture of Borel ideals. This enables us to prove theorems on the shape of the sectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function. Keywords: Borel sets, sectional matrices 1. Introduction There is a circle of ideas which goes back to the last century and is related to the connections between Algebraic Geometry and Algebraic Combinatorics. The starting point is the study of systems of polynomial equations, whose intrinsic difficulty leads to attacking the problems by degenerating the algebraic objects involved. Already at the beginning of this century, Macaulay had pointed out the importance of monomial ideals, showing that attached to every ideal there are several monomial ideals with the same Hilbert function. He proved that, given the value of the Hilbert function of an ideal in degree d, there is a lower bound for its value in degree d + 1, [Mac27]. Moreover, the lower bound is sharp and attained by some special monomial ideals, the lex-segment ideals. The importance of monomial ideals grew during the century, and they turned out to become a central tool in the theory of Hilbert schemes since they can be viewed as the limit of more complicated algebraic structures. More recently the availability of computational techniques such as Gr6bner bases has strengthened the link and brought right to the foreground the role of monomial ideals, which are themselves combinatorial objects. Monomial ideals obtained via Gr6bner bases can be viewed as degenerations of families of ideals, but there is a drawback in this construction. It depends on the co- ordinates. However, in the 1970s Galligo proved a fundamental result [Ga174] which * The authors were partially supported by the Consiglio Nazionale delle Ricerche (CNR). 197