We study how an irreducible closed algebraic curve X embedded in CP^3 can be recovered using its projections from points onto embedded projective planes. The different embeddings are unknown. The only input is the defining equation of each projected curve. We show how both the embeddings and the curve in CP^3 can be recovered modulo some actions of the group of projective transformations of CP^3. For two projections, we show how in a generic situation, a characteristic matrix of the two embeddings can be recovered. We also establish the minimal number of irreducible algebraic curves required to compute this characteristic matrix up to a finite-fold ambiguity, as a function of their degree and genus. Then we use this matrix to recover the class of the couple of maps and as a consequence to recover the curve. Then we consider another problem. N projections, with known projections operators and N >> 1, are considered as an input and we want to recover the curve. The recovery can be done by linear computations in the dual space and in the Grassmannian of lines in CP^3. A closely related question is also considered. Each point of a finite closed subset of an irreducible algebraic curve, is projected onto a plane from a different point. The projections operators are known. We show when and how the recovery of the algebraic curve is possible, in function of the degree of the curve of minimal degree generated by the centers of projection. A second part is devoted to applications to computer vision. The results in this paper solve a long standing problem in computer vision that could not have been solved without algebraic-geometric methods.
In this paper, we prove that a smooth hyperbolic projective curve over a finite field can be recovered from L-functions associated to the Hilbert class field of the curve and its constant field extensions. As a consequence, we give a new proof of a result of Mochizuki and Tamagawa that two such curves with isomorphic fundamental groups are themselves isomorphic.
A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute …
A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. As an application, it is proved that the Hermitian curve in positive characteristic coincides with the Galois closures of projections of some plane curve from some two non-uniform points.
We study the degrees of generators of the ideal of a projected Veronese variety $v_2(\mathbb{P}^3)\subset \mathbb{P}^9$ to $\mathbb{P}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in $\mathbb{A}^4$, Cremona transforms of $\mathbb{P}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in $\mathbb{P}^6$.
In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.
Let $S \subset \R^{k + m}$ be a compact semi-algebraic set defined by a system of $\ell$ polynomial inequalities of degree at most 2. $ Let $π$ denote the standard projection from $\R^{k + m}$ onto $\R^m$. We prove that for any $q >0$, the sum of the first $q$ Betti numbers of $π(S)$ is bounded by $(k + m)^{O(q\ell)}.$ We also present an algorithm for computing the the first $q$ Betti numbers of $π(S)$, whose complexity is $ (k+m)^{2^{O(q\ell)}}.$ For fixed $q$ and $\ell$, both the bounds are polynomial in $k+m$.
We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).
We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg's perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups, recovering, in particular, a rigidity theorem of Seitz.
Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space
We prove that two algebraic embeddings of a smooth variety $X$ in $\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\mathbb{C}^m$.
We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to alg…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work of the authors on projection-based tropical bases, we characterize algebraic properties of the relevant ideals and provide a characterization of the dual subdivision (as a subdivision of a fiber polytope). This dual subdivision naturally leads to the issue of self-intersections of a tropical variety under projections. For the case of curves, we provide some …
Given a $C^2$ semi-algebraic mapping $F: \mathbb{R}^N \rightarrow \mathbb{R}^p,$ we consider its restriction to $W\hookrightarrow \mathbb{R^{N}}$ an embedded closed semi-algebraic manifold of dimension $n-1\geq p\geq 2$ and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection $\frac{F}{\Vert F \Vert}:W\setminus F^{-1}(0)\to S^{p-1}$. Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering $W$ as spheres of small and big radii…
We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a generically finite dominant rational map: a congruence of normal lines to the rational surface. Then, an in-depth study of certain syzygy modules associated to such a congruence is presented and applied to build elimination matrices that provide universal representations of its finite fibers, under some genericity assumptions. These matrices depen…
With every smooth, projective algebraic curve $\tilde{C}$ with involution $σ:\tilde{C}\longrightarrow \tilde{C}$ without fixed points is associated the Prym data which consists of the Prym variety $P:=(1-σ)J(\tilde{C})$ with principal polarization $Ξ$ such that $2Ξ$ is algebraically equivalent to the restriction on $P$ of the canonical polarization $Θ$ of the Jacobian $J(\tilde{C})$. In contrast to the classical Torelli theorem the Prym data does not always determine uniquely the pair $(\tilde{C},σ)$ up to isomorphism. In this paper we introduce an extension of the Prym data as follows. We con…
We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines in $P^3$ . By using some well known material from algebraic geometry, we open new branches to computational vision. We introduce algebraic curves embedded in $P^3$ as the building blocks from which the tensor of a couple of cameras (projections) can be computed. In the process we address dimensional issues and as a result establish the minimal number of alge…
We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{gl}(k,{\mathbb C})$ correspond to algebraic curves $C$ of genus $(k-1)^2$, equipped with a flat projection $π:C\to{\mathbb P}^1$ of degree $k$, and an antiholomorphic involution $σ:C\to C$ covering the antipodal map on ${\mathbb P}^1$.
A code over a finite alphabet is called locally recoverable (LRC code) if every symbol in the encoding is a function of a small number (at most r) other symbols. A family of linear LRC codes that generalize the classic construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg. In this paper we extend this construction to codes on algebraic curves. We give a general construction of LRC codes on curves and compute some examples, including asymptotically good families of codes derived from the Garcia- Stichtenoth towers. The local recovery procedure is performed …
A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the structure sheaf of S, which is shown to admit a certain length 1 symmetrical locally free resolution. The structure theorem gives necessary and sufficient conditions for such a length 1 locally free symmetrical resolution of a sheaf F on P^3 in order that $Spec (mathcal F)$ yield a canonically projected surface of general type. The result was found in 1983 by the…
Nagoya University（名古屋大学）；Korea Institute for Advanced Study（韩国高等研究院）；The University of Tokyo（东京大学）
We discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky [HS09].
