Many classical problems in algebraic geometry have regained interest when techniques from differential geometry were introduced to study them. The modern foundations for this approach has been given by Griffiths and Harris in [6], who obtained in this way several classical and new results in algebraic geometry. More recently, this idea has been successfully followed by McCrory, Shifrin and Varley in [12] and [13] to study differential properties of hypersurfaces in P 3 and P 4 . In fact these two papers have greatly influenced the present work.In this spirit, the subject of this paper is the systematic study of focal surfaces of smooth congruences of lines in P 3 . This is indeed a clear example of a topic of differential nature in algebraic geometry. The study of such congruences has been very popular among classical algebraic geometers one century ago. Especially Fano has given many important contributions to this field. An essential ingredient in his work has been the focal surface of the congruence. This point of view has been retaken by modern algebraic geometers, such as Verra and Goldstein, and also by Ciliberto and Sernesi in higher dimension.What we find amazing in the papers by the classics is how much information they were able to provide about the focal surface of the known examples of congruences, in particular about its singular locus (and more especially about fundamental points). They seemed to have in mind some numerical relations that they never formulated explicitly. And even nowadays such kind of relations would require deep modern techniques, like multiple-point theory, but also this powerful machinery is not a priori enough since some generality conditions need to be satisfied.As a sample of this, the degree and class of the focal surface -the only invariants easy to compute-can be derived immediately from the Riemann-Hurwitz formula. However these invariants, even in the easiest examples (see Example 2.4 or Remarks after Corollary 4.7) seem to be wrong at a first glance. This is due to the existence of extra components of the focal surface or to the possibility that the focal surface counts with multiplicity, although this was never mentioned explicitly by the classics. Even in [5], these possibilities seem not to have been considered.The starting point of this work was to understand how the classics predicted the number of fundamental points of a congruence. We only know of one formula in the literature involving this number, which is however wrong (see Example 1.15 and the remark afterwards). So our first goal was to use modern techniques in order to rigorously obtain some of the classical results on the topic. Specifically, by regarding the focal surface of a congruence as a scheme, we reobtain its invariants (degree, class, class of its hyperplane 1 section, sectional genus, and degrees of the nodal and cuspidal curves) and give them a precise sense.We also restrict our attention to congruences of bisecants to a curve, or flexes to a surface (since they are special cases in the work by...

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CMF therapy apparently has no effect on REE either acutely or during a 6-month-period; the increased REE observed in the long-term is likely due to the concomitant increase in FFM. The lack of evidence of sarcopenic obesity, at variance with previous literature, is likely due to different patient selection.

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In Computer Vision, images of dynamic or segmented scenes are modeled as linear projections from P k to P 2. The reconstruction problem consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a variety in P k containing the objects for which the reconstruction fails. In this paper, we deal with projections both of points from P 4 → P 2 and of lines from P 3 → P 2. In both cases, we consider 3 projections, minimal number for a uniquely determined reconstruction. In the case of projections of points, we declinate the Grassmann tensors introduced in [21] in our context, and we use them to compute the equations of the critical locus. Then, given the ideal that defines this locus, we prove that, in the general case, it defines a Bordiga surface, or a scheme in the same irreducible component of the associated Hilbert scheme. Furthermore, we prove that every Bordiga surface is actually the critical locus for the reconstruction for suitable projections. In the case of projections of lines, we compute the defining ideal of the critical locus, that is the union of 3 α-planes and a line congruence of bi-degree (3, 6) and sectional genus 5 in the Grassmannian G(1, 3) ⊂ P 5. This last surface is biregular to a Bordiga surface [40]. We use this fact to link the two reconstruction problems by showing how to compute the projections of one of the two settings, given the projections of the other one. The link is effective, in the sense that we describe an algorithm to compute the projection matrices.