Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor is a necessary and sufficient condition in order that could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

     

Plane polar Cremona maps of arbitrarily large degree in positive characteristic

Geometriae Dedicata - A result of I.V. Dolgachev states that the complex homaloidal polynomials in three variables, i.e. the complex homogeneous polynomials whose polar map is birational, are of...     

Interpolation from spaces spanned by monomials

This is an extension and emendation of recent results on the use of Gauss elimination in multivariate polynomial interpolation and, in particular, ideal interpolation. Dedicated to Mariano Gasca on the occasion of his sixtieth birthday     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Stable intersection of affine Cantor sets defined by two expanding maps

It is shown that in the context of affine Cantor sets with two increasing maps, the arithmetic sum of both of its elements is a Cantor set otherwise, it is a closure of countable union of nontrivial intervals. Also, a new family of pairs of affine Cantor sets is introduced such that each element of it has stable intersection. At the end, pairs of affine Cantor sets are characterized such that the sum of elements of each pair is a closed interval.     

Extension of multilinear maps defined on subspaces

We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product of the entire spaces. We show that the ℓ _p spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2.     

Luigi Cremona

Ohne Zusammenfassung     

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofx ^k by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann ² and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p _k,n , which can be identified with a certain probability. The numberp _k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.     

Toric rings generated by special stable sets of monomials

In this paper we consider some subalgebras of the d-th Veronese subring of a polynomial ring, generated by stable subsets of monomials. We prove that these algebras are Koszul, showing that the presentation ideals have Gröbner bases of quadrics with respect to suitable term orders. Since the initial monomials of the elements of these Gröbner bases are square- free, it follows by a result of STURMFELS [S, 13.15], that the algebras under consideration are normal, and thus Cohen-Macaulay.     

Metric differentiability of Lipschitz maps defined on Wiener spaces

This note is devoted to the differentiability properties of -Lipschitz maps defined on abstract Wiener spaces and with values in metric spaces (Y,d _Y ). We prove the existence γ-a.e. of local seminorms in the Cameron-Martin space which allow to compute the metric derivative of the map. Second Author partially supported by DGES (Spain) MTM2006-03531.     

Tightness of certain monomials

For a quantized enveloping algebra of finite type, one can associate a natural monomial to a dominant weight. We show that these monomials for types A5 and D4 are semitight(i.e., a Z-linear combination of elements in the canonical basis) by a direct calculation.     

On infinitesimal and local rigidity of harmonic maps between spheres defined by spherical harmonics

Sunto Si studia la deformabilità infinitesimale e locale di mappe armoniche in sfere dimostrando che le immersioni minime standard f: S² Sⁿ (in particolare, la superficie di Veronese) sono localmente rigide. Si dà un esempio in cui la rigidità locale non implica la rigidità infinitesimale.

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During the preparation of this paper the second-named author was supported by the C.N.R. and enjoyed the hospitality of the Università di Cagliari, Istituto Matematico     

Real points in the fibres of dominant maps defined overR

Letf: X → Y be a dominant map between real algebraic varieties of the same dimension. The object of this paper is the study of the map which sends every pointy ∈ Y to the number of real points inf ⁻¹ (y). We give a sufficient condition ony to ensure that the above map is locally constant iny. Moreover, we consider the subset ofY where this map is no more locally constant and we give the principal properties of this set.

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Sunto Siaf: X → Y un morfismo dominante tra varietà algebriche reali della stessa dimensione. In questo articolo si studia la mappa che associa ad ogni puntoy ∈ Y il numero di punti reali inf ⁻¹ (y); in particolare si dà una condizione sufficiente suy affinchè tale mappa sia costante in un suo intorno. Inoltre, si considera il sottoinsieme diY in cui tale mappa non è più localmente costante e si danno le principali proprietà di tale insieme.

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The work was written while both the authors were members of the G.N.S.A.G.A. of the Consiglio Nazionale delle Ricerche.