By Alberto Corso, Philippe Gimenez, Maria Vaz Pinto, Santiago Zarzuela

Filled with contributions from overseas specialists, Commutative Algebra: Geometric, Homological, Combinatorial, and Computational facets positive aspects new study effects that borrow equipment from neighboring fields resembling combinatorics, homological algebra, polyhedral geometry, symbolic computation, and topology. This ebook contains articles offered in the course of meetings held in Spain and Portugal in June, 2003. It incorporates a number of themes, together with blowup algebras, Castelnuovo-Mumford regularity, quintessential closure and normality, Koszul homology, liaison idea, multiplicities, polarization, and mark downs of beliefs. This entire quantity will stimulate additional examine within the box.

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It is well-known that I is a prime ideal defining a Cohen-Macaulay ring and that I has a linear resolution. One checks that I 2 does not have a linear resolution in characteristic 0. 4. It is interesting to note that the graded Betti numbers of I and J as well as those of I 2 and J 2 coincide. Is this just an accident? There might be some hidden relationship between the two ideals. ” We can ask whether J is an initial ideal or a specialization (or an initial ideal of a specialization) of the ideal I.

Regularity jumps for powers of ideals 29 Let 1 ≤ b ≤ a ≤ d − 1 and F = MA − NB a binomial of bidegree ((a − b)(d + 1), b) in H such that φ(A) > φ(B) in the lex-order. Denote by v = (v1 , v2 ), u = (u1 , u2 ) the exponents of M and N and by α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ) the exponents of A and B. We collect all the relations that hold by assumption: (i) (ii) (iii) (iv) (v) (vi) 1 ≤ b ≤ a ≤ d −1 α1 + α2 + α3 = β1 + β2 + β3 = b v1 + v2 = u1 + u2 = (a − b)(d + 1) v1 + dα1 + (d − 1)α3 = u1 + dβ1 + (d − 1)β3 v2 + dα2 + α3 = u2 + dβ2 + β3 α1 + α2 > β1 + β2 .

1. 8 Let 1 < a < b be integers. Define the ideal I = (y2 zb1 , y2 zb2 , xz1b−1 z2 ) + z1b−a(y1 za1 , y1 za2 , xz1a−1 z2 ) + z1 z2 (z1 , z2 )b−1 of the polynomial ring K[x, y1 , y2 , z1 , z2 ]. We expect that reg(I) = b + 1 and reg(I k ) − reg(I k−1 ) > (b + 1) if k = a or k = b. Acknowledgments The author wishes to thank the organizers of the Lisbon Conference on Commutative Algebra (Lisbon, June 2003) for the kind invitation and for their warm hospitality. Some parts of this research project were carried out while the author was visiting MSRI (Berkeley) within the frame of the Special Program 2002/03 on Commutative Algebra.