We develop some basic facts on deformations of exterior differential ideals on a smooth complex algebraic variety. With these tools we study deformations of several types of differential ideals, leading to several irreducible components of the corresponding moduli spaces

We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations $$A \rightarrow B$$ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and (what we shall call) the structure-semiotics duality (of which the syntax-semantics duality for propositional and predicate logic are particular cases). Whereas in classical algebraic geometry a certain kind of  rings  can be recovered by considering their representations with respect to a unique codomain B, Grothendieck’s theory of schemes permits to reconstruct general (commutative) rings by considering representations with respect to a category of codomains. The strategy to reconstruct the object from its representations remains the same in both frameworks: the elements of the ring A can be realized—by means of what we shall generally call Gelfand transform—as quantities on a topological space that parameterizes the relevant representations of A.  As we shall argue, important dualities in different areas of mathematics (e.g. Stone duality, Gelfand duality, Pontryagin duality, Galois-Grothendieck duality, etc.) can be understood as particular cases of this general pattern. In the wake of Majid’s analysis of the Pontryagin duality, we shall propose a Kantian-oriented interpretation of this pattern. We shall use this conceptual framework to argue that Grothendieck’s notion of functor of points can be understood as a “relativization of the a priori” (Friedman) that generalizes the relativization already conveyed by the notion of domain extension to more general variations of the corresponding (co)domains.

This article deals with the irreducible components of the space of codimension one foliations in a projective space defined by logarithmic forms of a certain degree. We study the geometry of the natural parametrization of the logarithmic components and we give a new proof of the stability of logarithmic foliations, obtaining also that these irreducible components are reduced.

In this note we analyse the Exceptional Component of the space of integrable forms of degree two, introduced by Cerveau-Lins Neto, in terms of the geometry of Veronese curves and classical invariant theory.

Let $\mathcal F(r, d)$ denote the moduli space of algebraic foliations of codimension one and degree $d$ in complex proyective space of dimension $r$. We show that $\mathcal F(r, d)$ may be represented as a certain linear section of a variety of complexes. From this fact we obtain information on the irreducible components of $\mathcal F(r, d)$.

A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain some elliptic quartic curve. We also compute the degree of the locus of quartic surfaces containing an elliptic quartic curve, a case not covered by that formula.

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space $\mathscr F_q(r, d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space $\mathbb P^r$, when $1\le q \le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.

We show that the set of singular holomorphic foliations on projective spaces with split tangent sheaf and good singular set is open in the space of holomorphic foliations. We also give a cohomological criterion for the rigidity of holomorphic foliations induced by group actions and prove the existence of rigid codimension one foliations of degree n - 1 on Pn for every n ≥ 3.

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic polynomial of F. Also, we revisit the known sufficient condition in terms of Hankel matrices.

This paper demonstrates that, for axial non-central optical systems, the equation of a 3D line can be estimated using only four points extracted from a single image of the line. This result, which is a direct consequence of the lack of vantage point, follows from a classic result in enumerative geometry: there are exactly two lines in 3-space which intersect four given lines in general position. We present a simple algorithm to reconstruct the equation of a 3D line from four image points. This algorithm is based on computing the Singular Value Decomposition (SVD) of the matrix of Plucker coordinates of the four corresponding rays. We evaluate the conditions for which the reconstruction fails, such as when the four rays are nearly coplanar. Preliminary experimental results using a spherical catadioptric camera are presented. We conclude by discussing the limitations imposed by poor calibration and numerical errors on the proposed reconstruction algorithm.

A logarithmic 1-form on CPn can be written asGRAPHICSwith (F) over cap (i)=(Pi(m)(0) F-j)/F-i for some homogeneous polynomials F-i of degree d(i) and constants lambda(i) is an element of C* such that Sigma lambda(i)d(i)=0. For general F-i, lambda(i), the singularities of omega consist of a schematic union of the codimension 2 subvarieties F-i=F-j=0 together with, possibly, finitely many isolated points. This is the case when all F-i are smooth and in general position. In this situation, we give a formula which prescribes the number of isolated singularities.

In this note we analyse the Exceptional Component of the space of integrable forms of degree two, introduced by Cerveau-Lins Neto, in terms of the geometry of Veronese curves and classical invariant theory.

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic polynomial of F. Also, we review another well-known sufficient condition.

Let X ⊂ P^r be a smooth algebraic curve in projective space, over an algebraically closed field of characteristic zero. For each m ∈ N, the m-flexes of X are defined as the points where the osculating hypersurface of degree m has higher contact than expected, and a hypersurface H ⊂ P^r is called a m-Hessian if it cuts X along its m-flexes. When X is a complete intersection, we give an expression for a (rational) m-Hessian as the Div (in the sense of Grothendieck-Knudsen-Mumford) of a complex of graded free modules naturally associated to X. The construction of this complex involves relating sheaves of differential operators on a scheme and a subscheme, and higher Euler sequences on projective space.

We consider a rigidity question for isotropic harmonic maps from a compact Riemann surface to a complex projective space. In the case of the projective plane, we prove that ridigity holds if the degree is small in relation to the genus. For a projective space of any dimension we obtain coarser results about rigidity and rigidity up to finitely many choices.

The present note is motivated by the following well known result about vector bundles on projective space.

Interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for different model-based computer vision tasks has increased in recent years, and although their properties make them a natural choice for object recognition and positioning applications, algebraic curve and surface fitting algorithms often suffer from instability problems. One of the main reasons for these problems is that, while the data sets are always bounded, the resulting algebraic curves or surfaces are, in most cases, unbounded. In this paper, the authors propose to constrain the polynomials to a family with bounded zero sets, and use only members of this family in the fitting process. For every even number d the authors introduce a new parameterized family of polynomials of degree d whose level sets are always bounded, in particular, its zero sets. This family has the same number of degrees of freedom as a general polynomial of the same degree. Three methods for fitting members of this polynomial family to measured data points are introduced. Experimental results of fitting curves to sets of points in R/sup 2/ and surfaces to sets of points in R/sup 3/ are presented.< >

Powell is always called with a limited number of variables. For example, if the fitting surface has MxN control points, the maximum number of variables sent to Powell is around 3*(N-4). Only the bad parts of the strips and the meridians are tuned by Pow-ell. So, in practice, the number of variables is far below 3*(N-4). The caps only have (N+5) variables, which is also low. • We reduce the weight of the internal energy implicitly as the iteration goes on, because we have more confidence in the fitting surface after each iteration. This way, the discontinuities of the data can be well preserved. • We use the Powell minimization routine which is more stable, robust, and accurate than the gradient descent approach. • Due to the independency among the caps and meridians , our algorithm could run in parallel. • This system is easy to control because there are only two global parameters to adjust. • The assumptions of (1) one underlying object only, (2) the availability of good initial guess, and (3) geometrically simple objects without deep cavities have been the weakness points of the deformable model algorithms. By applying multiple snakes simultaneously and Boolean operations, objects can be segmented into independent ones, and cavities can also be well handled. Our algorithm makes the deformable model much more versatile. We would like to upgrade all algorithms in this paper completely to 3D ones, and build a working system for both 2D and 3D. In addition, we would like to work out a better 3D surface representation which can handle multiple objects and objects more complicated than Ge-nus 0. more irregular, but they are still continuous B-spline curves. So the objects and the hole can still be correctly segmented. The third experiment, in Figure 10, is on 3D data which is composed of two separate genus 1 toruses. (a) shows the data points, (b) is the result (object A) after the first fit, which results in a dumbbell-like shape, (c) is the residual of the data points that are not accounted for by object A. They are from the inner parts of the two toruses, and (d) shows the bad parts of object A without data points nearby. They are from the two ends and middle of object A. (e) is the merger of points in (c) and (d). (f) is the fitting result (object B) to points in …

which sends s∧t to s dt−t ds. J. Wahl made the striking observation that if C is embeddable in a K3 surface then ΦL is not onto for L = ΩC ([W], Thm. 5.9); this raises the natural problem of studying the stratification of the moduli space of curves Mg by the rank of the Wahl map Φ(C) = ΦΩ1C . Roughly speaking, our main theorem says that the closure of the locus of curves of genus 10 which lie on a K3 is equal to the locus where Φ(C) fails to be surjective. In order to state the theorem precisely and explain what is special about the case of genus 10, we need to introduce some spaces. Let Fg be the moduli space of K3 surfaces with a polarization of genus g, Pg the union, over all S ∈ Fg of the linear series |OS(1)|. Let K be the closure of the image of the natural rational map μ : Pg →Mg. As the dimension of Pg is 19+g and the dimension ofMg is 3g− 3, one might naively expect μ to be dominant for g ≤ 10 and finite onto its image for g ≥ 11. These expectations hold for g ≤ 9 ([M], Thm. 6.1) and for odd g ≥ 11 and even g ≥ 20 ([M-M], Thm. 1), but for g = 10, Mukai showed that μ is not dominant ([M], Thm. 0.7). This exceptional behavior is due to the fact that the general K3 surface of genus 10 is a codimension 3 plane section of a certain 5-fold, so that when a curve lies on a general K3, it in fact lies on a 3-dimensional family of them. One of our first tasks is to show that K is a divisor when g = 10. Over the open subset M10 of M10 of curves without automorphisms we have the relative Wahl map; let W denote its degeneracy locus and W the closure of W in M10. It is a theorem of Ciliberto-Harris-Miranda [C-H-M] that W is a divisor (i.e. the Wahl map does not degenerate everywhere), and by Wahl’s theorem K ≤ W. Our result can then be stated as follows.