On the smooth locus of aligned Hilbert schemes: the k-secant lemma and the general projection theorem

Let X be a smooth, connected, dimension n, quasi-projective variety imbedded in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with multiplicities k_i at points x_i (possibly coinciding). The expected dimension of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that the lines carrying this locus do not fill up \PP_NComment: 17 pages, revised versio

Order 1 Congruences of Lines with smooth Fundamental Scheme

In this note we present a notion of fundamental scheme for Cohen-Macaulay, order I, irreducible congruences of lines. We show that such a congruence is formed by the k-secant lines to its fundamental scheme for a number k that we call the secant, index of the congruence. if the fundamental scheme X is a smooth connected variety in FN, then k = (N — l)/(c — 1) (where c is the codimension of X) and X comes equipped with a special tangency divisor cut out by a virtual hypersurface of degree k — 2 (to be precise, linearly equivalent to a section by an hypersurface of degree (k — 2) without being cut by one). This is explained in the main theorem of this paper. This theorem is followed by a complete classification of known locally Cohen-Macaulay order 1 congruences of lines with smooth fundamental scheme. To conclude we remark that according to Zak’s classification of Severi Varieties and Hartshome conjecture for low codimension varieties, this classification is complete

On the projective dimension and the unmixed part of three cubics

Let $R$ be a polynomial ring over a field in an unspecified number of variables. We prove that if $J \subset R$ is an ideal generated by three cubic forms, and the unmixed part of $J$ contains a quadric, then the projective dimension of $R/J$ is at most 4. To this end, we show that if $K \subset R$ is a three-generated ideal of height two and $L \subset R$ an ideal linked to the unmixed part of $K$, then the projective dimension of $R/K$ is bounded above by the projective dimension of $R/L$ plus one.Comment: 23 pages; to appear in Journal of Algebr

Certain minimal varieties are set-theoretic complete intersections

We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two

Dépistage automatique de la rétinopathie diabétique dans les images de fond d’oeil à l’aide de l’apprentissage profond

RÉSUMÉ : Le diabète est une maladie chronique qui touche plus de 400 millions d’adultes dans le monde. Cette maladie peut entraîner plusieurs complications au cours de la vie d’un malade. Une de ces complications est la rétinopathie diabétique. Il s’agit de la principale cause de cécité chez l’adulte. Cette maladie apparaît souvent sans symptômes, il est donc important pour les personnes atteintes de diabète d’effectuer des vérifications régulières chez un ophtalmologue. Cette vérification s’effectue par la prise d’images numériques de fond d’oeil du patient. Ces images sont ensuite examinées par un médecin afin de donner un diagnostic. Dans ce travail, il est question d’automatiser le diagnostic de la rétinopathie diabétique à l’aide des images numériques de fond d’oeil ainsi que l’apprentissage profond. En effet, les réseaux de neurones ont suscité ces dernières années un intérêt important dans différents domaines, notamment ceux du médical et de la vision par ordinateur. Les réseaux convolutifs permettent des applications tel que la classification ou la segmentation d’images. Ici, la classification correspond à classer les images selon la présence ou non de la rétinopathie diabétique dans les images et la segmentation correspond à extraire des régions d’intérêt, comme les vaisseaux sanguins par exemple.----------ABSTRACT : Diabetes is a chronic disease that currently concerns more than 400 millions of adults in the world. This disease can cause several complications during the life of a person. One of these complications is diabetic retinopathy. Being one of the leading cause of blindness in the working age population, this complication is serious and requires medical prevention. This disease often appear without any symptoms, meaning that regular examinations with an ophtalmologist are required to enable its detection and treatment. This work is about automating the diagnostic of diabetic retinopathy, with the use of digital fundus images and deep learning. Indeed, deep learning and neural networks have recently been used in several fields, such as medical applications or computer vision. Convolutional neural networks perform applications such as image classification or image segmentation really well. Here, classification means to label each image based on the presence or absence of diabetic retinopthy in the images and segmentation means to extract regions of interest in the image, such as blood vessels

Multiplicities and a dimension inequality for unmixed modules

We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely-generated module M such that R/ann(M) is quasi-unmixed and contains a field, and that p and q are prime ideals in the support of M such that p is analytically unramified, p+q is m-primary and e(M_p)=e(M). Then dim(R/p)+dim(R/q)\leq dim(M). We also prove a similar theorem in a special case of mixed characteristic. Finally, we provide several examples to explain our assumptions as well as an example of a noncatenary, local domain R with prime ideal p such that e(R_p)>e(R)=1

When does depth stabilize early on?

In this paper we study graded ideals I in a polynomial ring S such that the numerical function f(k)=depth(S/I^k) is constant. We show that, if (i) the Rees algebra of I is Cohen-Macaulay, (ii) the cohomological dimension of I is not larger than the projective dimension of S/I and (iii) the K-algebra generated by some generators of I is a direct summand of S, then f(k) is constant. When I is a square-free monomial ideal, the above criterion includes as special cases all the results of a recent paper by Herzog and Vladoiu. In this combinatorial setting there is a chance that the converse of the above fact holds true.Comment: The title has been changed and other minor changes have been done. The paper will appear in Journal of Algebr

Segre Classes on Smooth Projective Toric Varieties

We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.Comment: 19 pages, 1 figure, added references, corrected typos, minor text replacement