By Jurgen Herzog, Gaetana Restuccia

This paintings is predicated at the lectures provided on the foreign convention of Commutative Algebra and Algebraic Geometry held in Messina, Italy. It discusses advancements and advances in commutative algebra, algebraic geometry, and combinatorics - highlighting the speculation of projective schemes, the geometry of curves, determinantal and sturdy beliefs, and unfastened resolutions.

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With nk+i > (HI + k). To get statement ii), repeatedly use the equality: ( 6 "j) + (£) = (^J). ) be an O-sequence with an M-hole in u and M-saturated for t > 1. Denote cu = ft) + ... + (™fth) the u-binomial expansion of cu. ). Then: 11) Ci is M-saturated in u + t, t > 2. 12) If ft ^ 1, Ci is M-saturated also in u + 1. is) If ft = 1, Ci has an (M,ni)-hole in u + 1. Moreover, the (u + 1)binomial representation of clu+l = cu+\ + 1 ends with (*). ) is M-saturated, for k > u, except in u + 1 , if n\ / 1 and in u + i, where it has an (M, l)-hole; moreover the (u + £)-binomial representation of cu+t + t ends with (}).

5 Letx = ( x i , . . ,a;d) £ ft, withv(x) = (QI,... , a s ,ft,+i,... - #<*» -a, < n* < oo, / o r i = 1,... ,s, tfien ( * ! , . . , Z d X t ; 1 ' , . . , * ? • , ! , . . , 1 ) 6 fl. Proof. Every element x' — (x(,... ,x's,Q,... ,0) of ft, with v^x'j) > St, for i = 1,... ,s has a value in v(C), sox' £ C C R. In particular (xi,... , xs, 0,... ,0) € ft and so (EI ,... ,x d )-(xi,... ,x s ,0,... ,0) = (0,... ,0,o;s+1,... ,xd) € ft, and thus every element ( x [ , . . , xd) with vi(x'i) > 6i, for i = 1,...

M. it INTRODUCTION This paper is strictly linked to a previous one,[l], in which properties of a pro- jective, 0-dimensional, reduced scheme are studied by means of a "refinement" of its Hilbert function or, actually, of its first difference F ([2],[3]). The main tool, in this investigation, is a numerical sequence Mr ; rather surprisingly, the set of those MF'S, arising from all the possibles F 's, turns out to be in 1-1 correspondence with the set of the r-type vectors introduced in [4] for a different purpose; moreover, a sequence of numbers of interest in the description of Mr turns out to be a generalization, to higher dimension, of the sequence of numerical characters introduced in [5] for a 0-dimensional scheme in P2.

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