On the grade of some ideals

An additivity formula is obtained for the grade of an ideal in a residue ring R/I, where I is a perfect ideal. This result is then applied to compute the grade of ideals of linear (inhomogeneous) polynomials. Further results on the homological rigidity of the conormal module I/I² are pointed out. Finally, an elementary proof is given of a result of Buchsbaum concerning the grade of ideals of minors of a matrix.Partially supported a CNPq grantPartially supported by NSF and CNPq grants     

Krull dimension and integrality of symmetric algebras

For a finitely generated moduleE over the Noetherian ringR we consider formulas for the Krull dimension of the symmetric algebraS(E). A result of Huneke and Rossi is re-proved and an effective formula is derived that reads the dimension ofS(E) from a presentation ofE. They provide a first line of obstructions forS(E) to be an integral domain. For algebras of codimension at most four we give methods, including computer-assisted ones, to ascertain whetherS(E) is a Cohen-Macaulay domain.AMS 1980 Mathematics Subject Classification (1985 Revision). Primary 13H10; Secondary 13-04, 13C05, 13C15Partially supported by CNPq, BrazilPartially supported by the NSF     

The divisor class group of ordinary and symbolic blow-ups

Mathematische Zeitschrift -     

The analytic spread of codimension two monomial varieties

Let ${\rm} A=k[{u_{1}^{a_{1}}},{u_{2}^{a_{2}}},\dots,{u_{n}^{a_{n}}},{u_{1}^{c_{1}}} \dots {u_{n}^{c_{n}}},{u_{1}^{b_{1}}} \dots {u_{n}^{b_{n}}}]\ \subset k[{u_{1}}, \dots {u_{n}}],$ where, a_j, b_j, C_j ∈ ?, a_j > 0, (b_j, C_j) ≠ (0,0) for 1 ≤ j ≤ n, and, further ${\underline b}:=\ ({b_{1}}, \dots,{b_{n}})\ \not=\ 0 $ and ${\underline c}:=\ ({c_{1}}, \dots,{c_{n}})\ \not=\ 0 $ . The main result says that the defining ideal I ? m = (x₁,…, x_n, y, z) ? k[x₁,…, x_n, y, z] of the semigroup ring A has analytic spread ?(I_m) at most three.     

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones.