Renato Maggioni

Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeOQ. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ3. For almost all such schemes the graded Betti numbers in ℙ3 depend only on the Hilbert function ofX in ℙ3.

On the rao module of a curve lying on a smooth cubic surface in p3, II

We find that every reduced and irreducible curve C on a smooth cubic surface is k-Buchsbaum, with k equal to the diameter of the Rao module M(C). Moreover we determine the diameters of the submodules generated by all the elements with the same degree in a minimal set of generators of M(C).

On the Resolution of a Curve Lying on a Smooth Cubic Surface in ℙ 3

Let C be any reduced and irreducible curve lying on a smooth cubic surface in P 3 . In this paper we determine the graded Betti numbers of the ideal sheaf J C

ON THE MULTIPLICATIVE STRUCTURE OF THE RAO MODULE OF SPACE CURVES LYING ON A SMOOTH QUADRIC

ABSTRACT The multiplication by a form gives a linear map between homogeneous summands of the Rao module; the matrix of this map is determined with respect to suitable bases. The rank of this matrix is studied in connection with properties of ; a complete description is obtained when is a form of degree 1 or 2 and some more results are found in particular cases.

On the resolution of a curve lying on a smooth cubic surface in

Let C be any reduced and irreducible curve lying on a smooth cubic surface in P3 . In this paper we determine the graded Betti numbers of the ideal sheaf J£ . Introduction In this paper we continue the study of curves C on a smooth cubic surface S of P3 begun by the first author in [G and Gl]. We obtain a complete description of the graded Betti numbers of the ideal sheaf Jc of C in terms of the seven integers which describe C as a divisor on S. A consequence of our results is that the graded Betti numbers of a reduced and irreducible curve C, lying on a smooth cubic (or quadric) surface S, do not change within the same linear equivalence class, i.e., they are determined by the class of Pic S to which C belongs. Observe that the starting point of our argument is the knowledge of Pic S as the group freely generated on a suitable basis. We would like to bring the readers attention on the algorithm of Remark 4.7. This was the key tool which first allowed us to understand the behavior of generators and syzygies of a curve on S. As far as we know this kind of problem has been solved, in general, only for arithmetically Cohen-Macaulay curves in P3 : the characterization of the graded Betti numbers for these curves was done in [E], where the work begun in [PS] is completed. The content of the various sections is the following: in § 1 we review some basic facts about curves on a smooth cubic and connect the graded Betti numbers of a curve with its Hilbert function. In §2 we determine the number of minimal generators of the homogeneous ideal 1(C), for each degree n , using the intrinsic geometric properties of the curves on 5e. In the third and fourth sections we find the numbers of the first and second syzygies for each degree n , completing in this way the description of the graded Betti numbers of Jc. Particular attention is devoted to curves generated in Received by the editors July 18, 1988 and, in revised form, February 6, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 14C20; Secondary 14J25. Work done with the financial support of the Italian M.P.I.

Generators for the ideal of an integral curve lying on a smooth quartic surface

Let C be a reduced and irreducible curve lying on a smooth (nongeneral) quartic surface S ⊂ P3; we find the number of elements, for any degree, in a minimal set of generators for the homogeneous ideal of C. In particular, we study the curves lying on certain surfaces called ‘Mori quartics’.

Sulla permanenza degli anelli di krull e completamente integralmente chiusi rispetto a certi tipi di omomorfismi

SummaryIn this paper we extend the property of a ringA to be of finite real character to loc-étaleA-algebras and to loc-étale neighbourhoods by a suitable representation of these algebras. By this means we get some result on the property to be completely integrally closed for loc-ind-étaleA-algebras and henselizations. We conclude by a counter-example which shows that the properties to be Krull and completely integrally closed are not preserved in the henselization.