BELTRAMETTI , Mauro Carlo

We give a completely different, much shorter, proof of a substantial generalization of the main result from \cite{KS}. It states that embedded projective $n$-folds swept out by quadrics of dimension at least $\big[\frac n2\big] +2$ are either scrolls or hyperquadric fibrations, which are also Mori contractions.

The philosophy that ``a projective manifold is more special than any of its smooth hyperplane sections" was one of the classical principles of projective geometry. Lefschetz type results and related vanishing theorems were among the typically used techniques. We shall survey most of the problems, results and conjectures in this area, using...

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Let $Y$ be a smooth curve embedded in a complex projective manifold $X$ of dimension $n\geq 2$ with ample normal bundle $N_{Y|X}$. For every $p\geq 0$ let $\alpha_p$ denote the natural restriction maps $\Pic(X)\to\Pic(Y(p))$, where $Y(p)$ is the $p$-th infinitesimal neighbourhood of $Y$ in $X$. First one proves that for every $p\geq 1$ there is...

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Let Y be a submanifold of dimension y of a polarized complex manifold (X; A) of dimension k 2, with 1 y k−1. We define and study two positivity conditions on Y in (X; A), called Seshadri A-bigness and (a stronger one) Seshadri A-ampleness. In this way we get a natural generalization of the theory initiated by Paoletti in [Paoletti...

Let $(\sM,\sL)$ be a smooth $4$-dimensional variety polarized by a very ample line bundle $\sL$. Let $\sA$ be a smooth member of $|\sL|$. Assume that $\sA$ is a Fano threefold of index one, with $-K_\sA\cong \sH_\sA$ for some ample line bundle $\sH_\sA$ on $\sA$. Let $\sH$ be the line bundle on $\sM$ which extends $\sH_\sA$. Up to sporadic...

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Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold $X$ inducing a covering family on a submanifold $Y$ with ample normal bundle in $X$, the main results relate, under suitable...

The Hough transform is a standard pattern recognition technique introduced between the 1960s and the 1970s for the detection of straight lines, circles, and ellipses. Here we offer a mathematical foun- dation, based on algebraic-geometry arguments, of an extension of this approach to the automated recognition of rational cubic, quartic, and...

Let (X, L) be a smooth polarized variety of dimension n. Let A ∈ |L| be an effective irreducible divisor, and let Σ be the singular locus of A. We assume that Σ is a smooth subvariety of dimension k ≥ 2, and codimension c ≥ 3, consisting of non-degenerate quadratic singularities. We study positivity conditions for adjoint bundles KX +tL with t...

The Hough transform is a standard pattern recognition technique introduced between the 1960s and the 1970s for the detection of straight lines, circles, and ellipses with several applications including the detection of symmetries in images. Recently, based on algebraic geometry arguments, the procedure has been extended to the automated...

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Let $X$ be a normal Gorenstein complex projective variety. We introduce the Hilbert variety $V_X$ associated to the Hilbert polynomial $\chi(x_1 L_1,\ldots, x_\rho L_\rho)$, where $ L_1,\ldots, L_\rho$ is a basis of $\Pic X$, $\rho$ being the Picard number of $X$, and $x_1,\ldots,x_\rho$ are complex variables. After studying general properties...

Let $f$ and $g$ be complex polynomials of the same degree. We provide a new lower bound on the Euclidean distance of points belonging to their zero-loci in terms of Bombieri's norm. We also present a minimization of the Bombieri's norm of the difference $g-\lambda f$, for $\lambda\in \C^{^*}$. In the real case, we apply the results above in...

Let $M$ be a $5$-dimensional manifold polarized by a very ample line bundle $L$. We show that a smooth $A\in |L|$ cannot be a holomorphic $\pn 1$-bundle over a smooth projective threefold $Y$, unless $Y\cong\pn 3$ and $A\cong\pn 1 \times\pn 3$.

Let $(\sM,\sL)$ be a smooth $(n+1)$-dimensional variety polarized by an ample and spanned line bundle $\sL$. Let $A$ be a smooth member of $|\sL|$. Assume that $n\geq 4$ and that $(A,H_A)$ is a Mukai variety, i.e., $-K_A\approx (n-2)H_A$ for some ample line bundle $H_A$ on $A$. Let $H$ be the line bundle on $\sM$ which extends $H_A$. We show...

Let f and g be complex multivariate polynomials of the same degree. Extending Beauzamy's results which hold in the univariate case, we bound the Euclidean distance of points belonging to the zero-loci of f and g in terms of the Bombieri norm of the difference g−f. We also discuss real perturbations of real polynomials.

Let $\sM$ be a smooth complex projective variety and let $\sL$ be a line bundle on it. Rays-positive manifolds, namely pairs $(\sM,\sL)$ such that $\sL$ is numerically effective and $\sL\cdot R>0$ for all extremal rays $R$ on $\sM$, are studied. Several illustrative examples and some applications are provided. In particular, projective ...

Let $(\sM,\sL)$ be a polarized threefold of log-general type. The birationality of the bicanonical map of a smooth surface $S\in|\sL|$ is studied. This problem was previously considered and partially solved by the first and fourth author, who gave a satisfactory classification unless $h^1(\sO_\sM)=0$ and $p_g(S)=3,4,5$. This paper focuses on...

We develop a formal procedure for the automated recognition of rational and elliptic curves in medical and astronomical images. The procedure is based on the extension of the Hough transform concept to the definition of Hough transform of special classes of algebraic curves. We first introduce a catalogue of curves that satisfy the conditions...

In this paper we present a Hough Transform-based method for the detection of the spinal district in X-ray Computed Tomography (CT) images in order to build binary masks that can be applied to functional images to infer information on the metabolic activity of the spinal marrow. This kind of information may be of particular interest for the...

This book offers a wide-ranging introduction to algebraic geometry along classical lines. It consists of lectures on topics in classical algebraic geometry, including the basic properties of projective algebraic varieties, linear systems of hypersurfaces, algebraic curves (with special emphasis on rational curves), linear series on algebraic...

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Many bone shapes in the human skeleton are characterized by profiles that can be associated to equations of algebraic curves. Fixing the parameters in the curve equation, by means of a classical pattern recognition procedure like the Hough transform technique, it is then possible to associate an equation to a specific bone profile. However,...