Marc Chardin

Implicitizing rational hypersurfaces using approximation complexes

We describe an algorithm for implicitizing rational hypersurfaces with at most a finite number of base points, based on a technique described in Buse, Laurent, Jouanolou, Jean-Pierre [2003. On the closed image of a rational map and the implicitization problem. J. Algebra 265, 312-357], where implicit equations are obtained as determinants of certain graded parts of an approximation complex. We detail and improve this method by providing an in-depth study of the cohomology of such a complex. In both particular cases of interest of curve and surface implicitization we also present algorithms which involve only linear algebra routines.

Liaison and Castelnuovo-Mumford regularity

In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities. When applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals.

Torsion of the symmetric algebra and implicitization

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of the MacRaes invariant of a graded part of the symmetric algebra is optimal. Then we show that the extraneous factor that may appear in the process splits into a product a linear forms in the algebraic closure of the base field, each linear form being associated to a non complete intersection base point. Finally, we make a link between this method and a resultant computation for the case of rational plane curves and space surfaces.

Castelnuovo–Mumford regularity: examples of curves and surfaces

The behaviour of Castelnuovo-Mumford regularity under “geometric” transformations is not well understood. In this paper we are concerned with examples which will shed some light on certain questions concerning this behaviour. One simple question which was open (see e.g. [R]) is: May the regularity increase if we pass to the radical or remove embedded primes? By examples, we show that this happens. As a by-product we are also able to answer some related questions. In particular, we provide examples of licci ideals related to monomial curves in P (resp. in P) such that the regularity of their radical is essentially the square (resp. the cube) of that of the ideal. It is well known that the regularity cannot increase when points (embedded or not) are removed. Hence, in order to construct examples where on removing an embedded component the regularity increases, we have to consider surfaces. More surprisingly, we find an irreducible surface such that, after embedding a line into it, the depth of the coordinate ring increases! Another important concept to understand is the limit of validity of Kodaira type vanishing theorems. The Castelnuovo-Mumford regularity of the canonical module of a reduced curve is 2. An analogous result holds true for higher dimensional varieties with isolated singularities (in characteristic zero), thanks to Kodaira vanishing. As a consequence one can give bounds for CastelnuovoMumford regularity (see [CU]). In [Mum], Mumford proves that for an ample line bundle L on a normal surface S, H(S,OS ⊗ L−1) = 0. He remarks that this is false if S doesn’t satisfy S2 (i.e. S is not Cohen-Macaulay) and asks if the S2 condition is sufficient. The first counter-example was given in [AJ]. In this article, we show that counter-examples satisfying S1 give rise to counterexamples satisfying S2. We then provide monomial surfaces whose canonical module has large Castelnuovo-Mumford regularity, so that this vanishing fails. We give a simple proof to show that if S satisfies R1, then H1(S, ωS ⊗L) = 0

Hilbert Functions, Residual Intersections, and Residually S2 Ideals

AbstractLet R be a homogeneous ring over an infinite field, I⊂R a homogeneous ideal, and

The eventual stability of depth, associated primes and cohomology of a graded module

\mathfrak{a}

Applications of some properties of the canonical module in computational projective algebraic geometry

⊂I an ideal generated by s forms of degrees d1,...,ds so that codim(