Mauro C. Beltrametti

The Chow motive of the Godeaux surface

Let X be the Godeaux surface obtained as a quotient of the Fermat quintic in P3C under the appropriate action of Z/5. We show that its Chow motive h(X) splits as 1 ⊕ 9L ⊕ L where L = (P, [P × pt]) is the Lefschetz motive. This provides a purely motivic proof of the Bloch conjecture for X. Our results also give a motivic proof of the Bloch conjecture for those surfaces considered in [BKL], i.e. all surfaces with pg = 0 which are not of general type.

On k-Jet Ampleness

Let X be an n-dimensional projective manifold mapped into a projective space Ψ:X → ℙℂ. Let L be the pullback, Ψ*Oℙℂ(1), of the hyperplane section bundle. If Ψ is an embedding, L is said to be very ample. This is an intensively studied and well-understood concept. In this chapter we study a particular notion of higher-order embedding. We say that L is k-jet ample for a nonnegative integer k if, given any r integers k 1 , ., k r , such that \( k + 1 = \sum\nolimits_{i = 1}^r {{k_i}} \) and any r distinct points {x 1 ,. . ., x r } ⊂ X, the evaluation map

Profile Detection in Medical and Astronomical Images by Means of the Hough Transform of Special Classes of Curves

X \times \Gamma (L) \to L/L \otimes m_{{x_1}}^{{k_1}} \otimes ... \otimes m_{xr}^{{k_r}} \to 0

Almost vanishing polynomials and an application to the Hough transform

is surjective, where m xi . denotes the maximal ideal at x t . Note that L is spanned (respectively, very ample) if and only if L is 0-jet ample (respectively, 1-jet ample).

Algebraic Geometry: A Volume in Memory of Paolo Francia

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 foundation, based on algebraic-geometry arguments, of an extension of this approach to the automated recognition of rational cubic, quartic, and elliptic curves. The accuracy of this approach is tested against synthetic data and in the case of experimental observations provided by the NASA Solar Dynamics Observatory mission.

On threefolds with low sectional genus

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 to be automatically extracted from an image and the recognition algorithm, then we illustrate the power of this method to identify skeleton profiles in clinical X-ray tomography maps and front ends of solar eruptions in astronomical images provided by the NASA solar dynamics observatory satellite.

Spinal Canal and Spinal Marrow Segmentation by Means of the Hough Transform of Special Classes of Curves

Let X be a nonsingular complex projective surface and let D be an ample divisor on X such that the associated invertible sheaf is spanned by its global sections. We prove that D is 2-connected apart from a few cases we explicitly describe. We also provide a corresponding result for the 3-connectedness when D2⩾10 and for the 4-connectedness when D2⩾17 and D is very ample.

A PET/CT approach to spinal cord metabolism in amyotrophic lateral sclerosis

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.