Search

We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to $\mathbb{p}^3$ of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal curve, while in the second we start by reconstructing the rational normal scroll. In both instances we then reconstruct the correct projection to $\mathbb{p}^3$ of these surfaces by exploiting the information contained in the singularities of the apparent contour., 17 pages

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q). We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension Fqn of Fq.

In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter--Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors., Final version. To appear in Proceedings of the National Academy of Sciences of the United States of America (PNAS)

We present a new algorithm to decide whether two rational parametric curves are related by a projective transformation and detect all such projective equivalences. Given two rational curves, we derive a system of polynomial equations whose solutions define linear rational transformations of the parameter domain, such that each transformation corresponds to a projective equivalence between the two curves. The corresponding projective mapping is then found by solving a small linear system of equations. Furthermore we investigate the special cases of detecting affine equivalences and symmetries as well as polynomial input curves. The performance of the method is demonstrated by several numerical examples.

How can the projective invariant of the cubic curve approximating the river bed near its meander be calculated? A well-known approach uses the Weierstrass normal form. However, it is important to find this form by means of calculations tolerant to curve representation errors and, in particular, using calculations that do not require computation of tangent lines or inflection points. A new algorithm is proposed for calculation of the projective invariant of the cubic curve. This algorithm can be used to describe river meanders.

This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces.

This is an introduction, aimed at a general mathematical audience, to recent work of Aprodu, Farkas, Papadima, Raicu and Weyman. These authors established a long-standing folk conjecture concerning the equations defining the tangent developable surface of a rational normal curve. This in turn led to a new proof of a fundamental theorem of Voisin on the syzygies of a general canonical curve. The present note, which is the write-up of a talk given by the second author at the Current Events seminar at the 2019 JMM, surveys this circle of ideas.

Pascal's Theorem gives a synthetic geometric condition for six points $a,\ldots,f$ in $\mathbb{P}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap\overline{de}$, $\overline{af}\cap\overline{dc}$, $\overline{ef}\cap\overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in $\mathbb{P}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann-Cayley algebra the defining equations of the parameter space of $d+4$ ordered points in $\mathbb{P}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic., 17 pages, 1 figure. Final version. To appear in Bulletin of the London Mathematical Society

Let $X \subset \mathbb{P}^{n}$ be an unramified real curve with $X(\mathbb{R}) \neq \emptyset$. If $n \geq 3$ is odd, Huisman conjectures that $X$ is an $M$-curve and that every branch of $X(\mathbb{R})$ is a pseudo-line. If $n \geq 4$ is even, he conjectures that $X$ is a rational normal curve or a twisted form of a such. We disprove the first conjecture by giving a family of counterexamples. We remark that the second conjecture follows for generic curves of odd degree from the formula enumerating the number of complex inflection points., Comment: 9 pages, 2 figures

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

In this paper we study an instance of projective Reed-Muller type codes, i.e., codes obtained by the evaluation of homogeneous polynomials of a fixed degree in the points of a projective variety. In our case the variety is an important example of a determinantal variety, namely the projective surface known as rational normal scroll, defined over a finite field, which is the basic underlining algebraic structure of this work. We determine the dimension and a lower bound for the minimum distance of the codes, and in many cases we also find the exact value of the minimum distance. To obtain the results we use some methods from Grobner bases theory.

The zeroth stable $$ \mathbb{A} $$ 1-homotopy group of a smooth projective variety is computed. This group is identified with the group of oriented 0-cycles on the variety. The proof heavily exploits properties of strictly homotopy invariant sheaves. Bibliography: 7 titles.

We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as a disjoint union of locally closed subvarieties in the Grassmannian $\mathbb{G}(n-2,n)$., 12 pages, comments welcome

Geometries in higher dimensional spaces have many applications. We shall give a compilation of a few well-known examples here. The fact that some higher dimensional geometries can be found within some lower dimensional geometries makes them even more interesting. At hand of some familiar examples, we shall see what these concepts in geometry can do for us. In the beginning, the meaning of dimension will be clarified and an agreement is reached about what is higher dimensional. A few words will be said about the relations and interplay between models of various geometries. To the concept of model spaces a major part of this contribution will be dedicated to. A full section is dedicated to the applications of higher dimensional geometries.

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve., minor edits, 42 pages, to appear in Invent. Math

Let $X(\RR)$ be a geometrically connected variety defined over $\RR$ and such that the set of all its (also complex) points $X(\CC)$ is non-degenerate. We introduce the notion of \emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set of points of $X(\CC)$ such that $P\in \langle S \rangle$ that is stable under conjugation. Any set evincing the admissible rank can be equipped with a \emph{label} keeping track of the number of its complex and real points. We show that in the case of generic identifiability there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if $X$ is a rational normal curve than there always exists a label for the generic element. We present two examples in which either the label doesn't exists or the admissible rank is strictly bigger than the usual complex rank., 12 pages, Comments welcome

The basis of H-Bezier curves of degree n is 1 , t , ? , t n - 2 , sinh α t and cosh α t , for t ? 0 , 1 ] . We find the limit curve of H-Bezier curves of degree n as a parameter α goes to ∞ , which is the Bezier curve of degree n - 2 , and prove it using mathematical induction and special properties of H-basis functions. We also compare it to the limit curve of rational Bezier curves of degree n in standard form with the same weight w as it goes to ∞ , which is the rational Bezier curve of degree n - 2 .

We investigate the relationship between rational plane curves and the envelopes defined by the syzygies of their parameterizations.

The parameter space of $n$ ordered points in projective $d$-space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in $(\mathbb{P}^d)^n$. The resulting variety was used to study the birational geometry of the moduli space $\overline{\mathrm{M}}_{0,n}$ of $n$-tuples of points in $\mathbb{P}^1$. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely $d=2$, we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, $d=3$, we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For $d \ge 4$ we conjecture a similar situation and prove partial results in this direction., Comment: 28 pages. Minor correction. We removed the erroneous Lemma 4.7 in the previous version, but the remaining results are valid

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting type of their restricted tangent bundles, compute their normal bundles and show how to construct these curves as suitable projections of a rational normal curve., A new Lemma 8 added, to the purpose of clarifying the proof of Proposition 7 and for future reference

Let $X_0$ be a smooth hypersurface (assumed not to be generic) in projective space $\mathbf {P}^n$, $n\geq 4$, over complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in deformations of the pair $C_0$ and $X_0$. In this paper, we prove that, if the first order deformations of the pair exist along each deformation of the hypersurface $X_0$, then $\deg (C_0)$ cannot be in the range \[ \bigg ( m\frac {2\deg (X_0)+1}{\deg (X_0)+1}, \frac {2+m(n-2)}{2n-\deg (X_0)-1}\bigg ), \] where $m$ is any non negative integer less than \[ \dim (H^0(\mathcal {O}_{\mathbf {P}^n}(1))|_{C_0} )-1. \]

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(\lambda_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $\omega$ as the monodromy operator. We provide a local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$., Comment: 33 pages, 12 figures including 2 color figures; to appear in the Journal of Algebraic Combinatorics

A distribution whose normalization constant is an A-hypergeometric polynomial is called an A-hypergeometric distribution. Such a distribution is in turn a generalization of the generalized hypergeometric distribution on the contingency tables with fixed marginal sums. In this paper, we will see that an A-hypergeometric distribution with a homogeneous matrix of two rows, especially, that associated with the rational normal curve, appears in inferences involving exchangeable partition structures. An exact sampling algorithm is presented for the general (any number of rows) A-hypergeometric distributions. Then, the maximum likelihood estimation of the A-hypergeometric distribution associated with the rational normal curve, which is an algebraic exponential family, is discussed. The information geometry of the Newton polytope is useful for analyzing the full and the curved exponential family. Algebraic methods are provided for evaluating the A-hypergeometric polynomials., Comment: 36 pages, 2 figures

We compute the facets of the effective and movable cones of divisors on the blow-up of at n + 3 points in general position. Given any linear system of hypersurfaces of based at n + 3 multiple points in general position, we prove that the secant varieties to the rational normal curve of degree n passing through the points, as well as their joins with linear subspaces spanned by some of the points, are cycles of the base locus and we compute their multiplicity. We conjecture that a linear system with n + 3 points is linearly special only if it contains such subvarieties in the base locus and we give a new formula for the expected dimension.

We consider the parameterization ${\mathbf{f}}=(f_0,f_1,f_2)$ of a plane rational curve $C$ of degree $n$, and we want to study the singularities of $C$ via such parameterization. We do this by using the projection from the rational normal curve $C_n\subset \mathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. In particular, we define via ${\mathbf{f}}$ certain 0-dimensional schemes $X_k\subset \mathbb{P}^k$, $2\leq k\leq (n-1)$, which encode all information on the singularities of multiplicity $\geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow to get info about the singularities from such schemes., Comment: 23 pages, 4 algorithms. Conjecture 4.4 in v1, now is Proposition 4.4. Accepted for the publication in JSC

Cryptographic algorithms in smart cards and other constrained environments increasingly rely on Elliptic Curves and thus it is desirable to have fast algorithms for elliptic curve arithmetic. In this paper, we provide (i) faster differential addition formulae for elliptic curve arithmetic on Generalized Edwards’ Curves improving upon the currently known formulae in the literature, proposed by Justus and Loebenberger at IWSEC 2010, (ii) more efficient affine differential addition formulae for a new model of Binary Edwards Curves proposed by Wu, Tang and Feng at INDOCRYPT 2012 and (iii) an algorithm for point doubling on Twisted Edwards Curves with a smaller footprint when the implementation is desired to work across Homogeneous Projective, Inverted and Extended Homogeneous Projective Coordinates.

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field F q \mathbf {F}_q . This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.

We introduce and study projective varieties \(X^{r+1} \subset \mathbb{P}^{N}\) of dimension r + 1 which are n-covered by irreducible curves of degree δ ≥ n − 1 ≥ 1, that is, varieties such that through n-general points there passes an irreducible curve of degree δ contained in X, denoted by Xr+1(n, δ). We present the sharp Pirio–Trepreau bound for the embedding dimension N in terms of r, n, δ in Theorem 6.2.3, which is obtained geometrically via the iteration of projections from general osculating spaces to \(X^{r+1}(n,\delta ) \subset \mathbb{P}^{N}\), determined by the irreducible curves of degree δ which n-cover the variety. The varieties extremal for the previous bound are subject to even stronger restrictions—e.g. they are rational and through n general points there passes a unique rational normal curve of degree δ, see Theorem 6.3.2 and Theorem 6.3.3. The main result of Pirio and Trepreau (Bull Soc Math Fr 141:131–196, 2013) ensures that the examples of Castelnuovo type are the only extremal varieties except possibly when n > 2, r > 1 and δ = 2n − 3. The first open case, that is, the classification of extremal varieties \(X = X^{r+1}(3,3) \subset \mathbb{P}^{2r+3}\) not of Castelnuovo type, is considered in Sect. 6.4, where it is proved that these varieties are in one-to-one correspondence, modulo projective transformations, with quadro-quadric Cremona transformations on \(\mathbb{P}^{r}\), Pirio and Russo (Commentarii Math Helv 88:715–756, 2013, Theorem 5.2) and Theorem 6.4.5 here. We deduce from this that a quadro-quadric Cremona transformation is, modulo projective transformations acting on the domain and on the codomain, an involution, see Corollary 6.4.6. We obtain the classification of smooth extremal varieties \(X^{r+1}(3,3) \subset \mathbb{P}^{2r+3}\) showing that there are two infinite series: smooth rational normal scrolls and \(\mathbb{P}^{1} \times Q^{r}\) Segre embedded; and four isolated examples appearing for r = 5, 8, 14 and 26 whose \(\mathcal{L}_{x} \subset \mathbb{P}^{r}\) is one the four Severi varieties. In Sect. 6.5 we include a self-contained presentation of the basics of the theory of power associative algebras and of their subclass of Jordan algebras, generalizing to this setting the usual Laplace formulas for inversion of a square matrix. In Theorem 6.5.22 we recall that every quadro-quadric Cremona transformation is linearly equivalent to the cofactor or adjoint map of a suitable rank three Jordan algebra, see loc. cit. for details. We end the chapter by surveying the recent results in Pirio and Russo (J. Reine Angew. Math, 2014, to appear) showing that extremal Xr+1(3, 3) and quadro-quadric Cremona transformations are also in bijection with the isotopy classes of rank three complex Jordan algebras, see Sect. 6.6 for precise formulations of these equivalences leading to the so-called XJC-correspondence, defined in Pirio and Russo (J. Reine Angew. Math, 2014, to appear).

It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. We prove a generalization of this to higher dimensional varieties, showing that smooth varieties of minimal degree can be interpolated through points and linear spaces., Comment: 22 pages, 7 figures

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real -- in this case, for zero-dimensional Schubert problems, the solutions are "as real as possible". Recent work by Speyer has extended the theory to the moduli space $\overline{M_{0,r}}$, allowing the points to collide. These give rise to smooth covers of $\overline{M_{0,r}}(\mathbb{R})$, with structure and monodromy described by Young tableaux and jeu de taquin. In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{M_{0,r}}$. Their (real) geometry turns out to be described by orbits of Sch\"{u}tzenberger promotion and a related operation involving tableau evacuation. Over $M_{0,r}$, our results show that the real points of the solution curves are smooth. We also find a new identity involving `first-order' K-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof., Comment: 32 pages, 10 pages

© 2014 Elsevier B.V. Allrightsreserved. A swept surface is generated from a profile curve and a sweep curve by employing the latter to define a continuous family of transformations of the former. By using polynomial or rational curves, and specifying the homogeneous coordinates of the swept surface as bilinear forms in the profile and sweep curve homogeneous coordinates, the outcome is guaranteed to be a rational surface compatible with the prevailing data types of CAD systems. However, this approach does not accommodate many geometrically intuitive sweep operations based on differential or integral properties of the sweep curve - such as the parametric speed, tangent, normal, curvature, arc length, and offset curves - since they do not ordinarily have a rational dependence on the curve parameter. The use of Pythagorean-hodograph (PH) sweep curves surmounts this limitation, and thus makes possible a much richer spectrum of rational swept surface types. A number of representative examples are used to illustrate the diversity of these novel swept surface forms - including the oriented-translation sweep, offset-translation sweep, generalized conical sweep, and oriented-involute sweep. In many cases of practical interest, these forms also have rational offset surfaces. Considerations related to the automated CNC machining of these surfaces, using only their high-level procedural definitions, are also briefly discussed.

We describe a relation between the invariants of $n$ ordered points in $P^d$ and of points contained in a union of linear subspaces $P^{d1}\cup P^{d2} \subset P^d$. This yields an attaching map for GIT quotients parameterizing point configurations in these spaces, and we show that it respects the Segre product of the natural GIT polarizations. Associated to a configuration supported on a rational normal curve is a cyclic cover, and we show that if the branch points are weighted by the GIT linearization and the rational normal curve degenerates, then the admissible covers limit is a cyclic cover with weights as in this attaching map. We find that both GIT polarizations and the Hodge class for families of cyclic covers yield line bundles on $\bar{M}_{0,n}$ with functorial restriction to the boundary. We introduce a notion of divisorial factorization, abstracting an axiom from rational conformal field theory, to encode this property and show that it determines the isomorphism class of these line bundles. As an application, we obtain a unified, geometric proof of two recent results on conformal block bundles, one by Fedorchuk and one by Gibney and the second author., Comment: 17 pages, 3 figures

Canonical surfaces with geometric genus four are studied assuming that the image of the canonical map is a normal surface in projective 3-space. It is shown that the degree of the image does not exceed [Formula: see text]. Furthermore, normal canonical sextics surfaces are explicitly constructed, extending a former example due to Zariski.