Francesco Russo

Bio: Francesco Russo is an academic researcher from University of Catania. The author has contributed to research in topics: Projective variety & Projective space. The author has an hindex of 19, co-authored 49 publications receiving 1028 citations. Previous affiliations of Francesco Russo include Federal University of Pernambuco & Romanian Academy.

On the Geometry of Some Special Projective Varieties

Abstract: Preface.-Introduction.- 1.Tangent cones, tangent spaces, tangent stars secant, tangent and tangent star varieties to an algebraic variety.- 2.Basics of Deformation Theory of Rational Curves on Projective Varieties.- 3.Fulton-Hansen Connectedness Theorem, Scorza Lemma and their applications to projective geometry.- 4.Local quadratic entry locus manifolds and conic connected manifolds.- 5.Hartshorne Conjectures and Severi varieties.- 6.Varieties n-covered by curves of a fixed degree and the XJC.- 7. Hypersurfaces with vanishing hessian.-Bibliography

On Birational Maps and Jacobian Matrices

Abstract: One is concerned with Cremona-like transformations, i.e., rational maps from ℙn to ℙm that are birational onto the image Y ⊂ ℙm and, moreover, the inverse map from Y to ℙn lifts to ℙm. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.

Varieties with quadratic entry locus, I

Abstract: We introduce and study (L)QEL-manifolds \({X \subset \mathbb P^N}\) of type δ, a class of projective varieties whose extrinsic and intrinsic geometry is very rich, especially when δ > 0. We prove a strong Divisibility Property for LQEL-manifolds of type δ ≥ 3, allowing the classification of those of type \({\delta \geq \frac{dim(X)}{2}}\) . In particular we obtain a new and very short proof that Severi varieties have dimension 2,4, 8 or 16 and also an almost self-contained proof of their classification due to Zak. We also provide the classification of special Cremona transformations of type (2,3) and (2,5).

Classical Algebraic Geometry: A Modern View

Abstract: Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

On the concept of k-secant order of a variety

Abstract: For a variety X of dimension n in ${\mathbb P}^r,\ r\geq n(k+1)+k$ , the k th secant order of X is the number $\mu_k(X)$ of $(k+1)$ -secant k -spaces passing through a general point of the k th secant variety. We show that, if $r>n(k+1)+k$ , then $\mu_k(X)=1$ unless X is k -weakly defective. Furthermore we give a complete classification of surfaces $X\subset{\mathbb P}^r,\ r>3k+2$ , for which $\mu_k(X)>1$ .

Decomposition of homogeneous polynomials with low rank

Abstract: Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the s-th secant variety of the d-uple Veronese embedding of $${\mathbb{P}^m}$$ into $${\mathbb{P}^{\tiny\left(\begin{array}{c}{\rm m+d} \\ {\rm d}\end{array}\right)-1}}$$ but that its minimal decomposition as a sum of d-th powers of linear forms M 1, . . . , M r is $${F=M_1^d+\cdots + M_r^d}$$ with r > s. We show that if s + r ≤ 2d + 1 then such a decomposition of F can be split in two parts: one of them is made by linear forms that can be written using only two variables, the other part is uniquely determined once one has fixed the first part. We also obtain a uniqueness theorem for the minimal decomposition of F if r is at most d and a mild condition is satisfied.