Edoardo Ballico

On smooth subcanonical varieties of codimension 2 inP n ,n⩾4

SummaryWe study subcanonical codimension 2 subvarieties ofPn, n ⩾ 4, using as our main tool the rank 2 vector bundle canonically associated to them. With this method we prove first that every smooth canonical surface inP4 is a complete intersection. Next we study smooth varieties of codimension 2inPn,n⩾6; it is well known that all of them are subcanonical and R. Hartshorne conjectured that they are always complete intersections, if n⩾7. We prove this conjecture in the particular case of a variety X for which the integer e such that Ωx=θx(e) is 0 or negative. This result, togheter with a strong result by Z. Ran, provides a quadratic bound for the degree of a non-complete intersection variety of codimension 2inPn, n⩾6.

Stratification of the fourth secant variety of Veronese variety via the symmetric rank

Abstract. If is a projective non-degenerate variety, the X-rank of a point is defined to be the minimum integer r such that P belongs to the span of r points of X. We describe the complete stratification of the fourth secant variety of any Veronese variety X via the X-rank. This result has an equivalent translation in terms either of symmetric tensors or of homogeneous polynomials. It allows to classify all the possible integers r that can occur in the minimal decomposition of a homogeneous polynomial of X-border rank (i.e. contained in the fourth secant variety) as a linear combination of powers of linear forms.

On the weak non-defectivity of Veronese embeddings of projective spaces ∗

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+x). Here we prove that the order x Veronese embedding ofPn is not weakly (k−1)-defective, i.e. for a general S⊃Pn such that #(S) = k+1 the projective space | I2S(x)| of all degree t hypersurfaces ofPn singular at each point of S has dimension (n/n+x)−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I2S(x)| has an ordinary double point at each P∈ S and Sing (F)=S.

The Horace Method for Error-Correcting Codes: Mathematics Subject Classification (2000)

Here we apply the so-called Horace method for zero-dimensional schemes to error-correcting codes on complete intersections. In particular, we obtain sharper estimates on the minimum distance.

Tensor ranks on tangent developable of Segre varieties

We describe the stratification by tensor rank of the points belonging to the tangent developable of any Segre variety. We give algorithms to compute the rank and a decomposition of a tensor belonging to the secant variety of lines of any Segre variety. We prove Comons conjecture on the rank of symmetric tensors for those tensors belonging to tangential varieties to Veronese varieties.

On the birational geometry of moduli spaces of pointed curves

Abstract We prove that the moduli space ℳ g,n of smooth curves of genus g with n marked points is rational for g = 6 and 1 ≤ n ≤ 8, and it is unirational for g = 8 and 1 ≤ n ≤ 11, g = 10 and 1 ≤ n ≤ 3, g = 12 and n = 1.

Hierarchical Secret Sharing in Ad Hoc Networks through Birkhoff Interpolation

Secret sharing schemes provide a natural way of addressing security issues in mobile ad hoc networks. This paper introduces a flexible framework for secure end-to-end transmission of confidential information which exploits multipath source routing and hierarchical shares distribution. Such a goal is achieved by designing a ideal, perfect, and eventually verifiable secret sharing scheme based on Birkhoff polynomial interpolation and by establishing suitable hierarchies among independent paths.

Globally generated vector bundles of rank 2 on a smooth quadric threefold

Abstract We investigate the existence of globally generated vector bundles of rank 2 with c 1 ≤ 3 on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q with c 1 = 3 is an odd instanton up to twist.

EXACT SEQUENCES OF SEMISTABLE VECTOR BUNDLES ON ALGEBRAIC CURVES

Let X be a smooth complex projective curve of genus g ≥ 1. If g ≥ 2, assume further that X is either bielliptic or with general moduli. Fix integers r, s, a, b with r > 1, s > 1 and as ≤ br. Here we prove the existence of an exact sequence 0 → H → E → Q → 0 of semistable vector bundles on X with rk(H) = r, rk(Q) = s, deg(H) = a and deg(Q) = b.

Curves of maximum genus in range A and stick-figures

In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than m, with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Floystad.