Mirella Manaresi

An axiomatic approach to the second theorem of Bertini

Abstract We give an axiomatic proof of the second theorem of Bertini and we show to which extent it can be applied to weak normality.

Sard and Bertini type theorems for complex spaces

SummaryWe prove that if X is a normal [reap. reduced, maximal] complex space and f: X→C is a holomorphic function, then f−1(c) is normal [resp. reduced, maximal] for all but countably many cεC. This Sard type theorem, together with a Bănicăs result on the fibers of a flat map, allows us to prove Bertini type theorems for reduced and normal complex spaces.

A length formula for the multiplicity of distinguished components of intersections

Abstract To an arbitrary ideal I in a local ring (A, m ) one can associate a multiplicity j(I,A) that generalizes the classical Hilbert–Samuel multiplicity of an m -primary ideal and which plays an important role in intersection theory. If the ideal is strongly Cohen–Macaulay in A and satisfies a suitable Artin–Nagata condition then our main result states that j(I,M) is given by the length of I/(x1,…,xd−1)+xdI where d≔ dim A and x1,…,xd are sufficiently generic elements of I. This generalizes the classical length formula for m -primary ideals in Cohen–Macaulay rings. Applying this to an hypersurface H in the affine space we show for instance that an irreducible component C of codimension c of the singular set of H appears in the self-intersection cycle Hc+1 with multiplicity e( jac H,C , O H,C ) , where jacH is the Jacobian ideal generated by the partial derivatives of a defining equation of H.

On the jumping conics of a semistable rank two vector bundle on P2

Stromme (see [S], (1.7)) introduced the notion of jumping conic of a normalized semistable rank two vector bundle E on ℙ2 and he remarked that the locus of jumping conics of E has codimension ≤3+2c1(E), with equality for general E.Here we introduce a concept of jumping conic of a semistable rank two vector bundle on ℙ2 (see (I.1)) by generalizing the notion of jumping line of the second kind introduced by Hulek in [H]. Our definition agrees with Strommes for c1=−1, but not for c1=0. In contrast with the case of jumping lines, where we have a different behaviour in the case c1 even or c1 odd, the set of jumping conics according to our definition is always a divisor (possibly empty) in the ℙ5 of all conics of ℙ2 (see th. (I.8)), whose degree depends on c2(E).

On the singularities of weakly normal varieties

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.

Analytic deviation of ideals and intersection theory of analytic spaces

In this note a stratification of analytic subspaces by analytic deviation of ideals is introduced and applied to define embedded components of an intersection (not necessarily proper) of complex analytic subspaces. Using the method of compact semi-analytic Stein neighbourhoods, a pointwise defined intersection multiplicity is proved to be constant along a dense Zariski open subset of such embedded components. For complex subspaces of a projective space one obtains precisely the distinguished varieties of intersection in the sense of Fulton and their intersection numbers.

On the self-intersection cycle of surfaces and some classical formulas for their secant varieties

Abstract We study the relation between certain local and global numerical invariants of a projective surface given by the Stückrad–Vogel self-intersection cycle. In the smooth case we find a relation between the degrees of the tangent, the secant and the first polar variety which generalizes classical relations between the so-called elementary invariants of surfaces. For singular surfaces this relation involves also the contributions of singularities to the self-intersection cycle. In order to make the formula really effective for singular surfaces, we need to describe the movable part of the self-intersection cycle on the singular locus. This is done explicitly for a particular class of non-normal del Pezzo surfaces.

Equimultiplicity and Equidimensionality of Normal Cones

The purpose of this paper is to give a new approach to some classical results on equimultiplicity. Our proofs are based on [FVo] in which there was described in a very precise way the behaviour of Hilbert function and multiplicities in an exact sequence and, in particular, under hyperplane sections. In [FOV], Section 1.1, this approach was used to give a simplified account to the classical theory of multiplicities including Rees’ theorem.

Fermat’s Last Theorem

This article generalizes and makes some additions to the method used in this demonstration theorem for exponents 3 and 5. In this regard, this paper presents a complete algebraic demonstration of Fermat’s Last Theorem.

SELF-INTERSECTIONS OF SURFACES AND WHITNEY STRATIFICATIONS

Let X be a surface in C n or P n and let CX(X £ X) be the normal cone to X in X £ X (diagonally embedded). For a point x 2 X, denote by g(x) := ex(CX(X £ X)) the multiplicity of CX(X £ X) at x. It is a former result of the authors that g(x) is the degree at x of the Stuckrad-Vogel cycle v(X;X) = C j(X;X;C)(C) of the self-intersection of X, that is, g(x) = C j(X;X;C)ex(C). We prove that the stratification of X by the multiplicity g(x) is a Whitney stratification, the canonical one if n = 3. The corresponding result for hypersurfaces in A n or P n , diagonally embedded in a multiple product with itself, was conjectured by L. van Gastel. This is also discussed, but remains open.