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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

The current study presents rational cubic trigonometric Bezier curve with two shape parameters, which is a novel technique for drawing free form curves. The proposed curve retains most of the geometric properties of conventional rational cubic Bezier but is flexible due to the presence of shape parameters. The shape of the curve can be adjusted locally by altering the values of shape parameters as well as the weights. Using the given end point curvatures, conditions on shape parameters and weights have been driven so that the curve always lies in the convex hull of its control point. Later, this curve is used to generate piecewise rational trigonometric curves stitched together using parametric and geometric Hermite continuity conditions which can facilitate a designer to generate a smooth composition of curves in a situation where single curve does not work. The piecewise curves thus constructed are used to draw the layout of certain English and Arabic alphabets as an application in font designing.

Rational Bezier curves provide a curve fitting tool and are widely used in Computer Aided Geometric Design, Computer Aided Design and Geometric Modeling. The injectivity (one-to-one property) of rational Bezier curve as a mapping function is equivalent to the curve without self-intersections. We present a geometric condition on the control polygon which is equivalent to the injectivity of rational Bezier curve with this control polygon for all possible choices of weights. The proof is based on the degree elevation and toric degeneration of rational Bezier curve.

We present a simple method for C-shaped G 2 Hermite interpolation by a rational cubic Bezier curve with conic precision. For the interpolating rational cubic Bezier curve, we derive its control points according to two conic Bezier curves, both matching the G 1 Hermite data and one end curvature of the given G 2 Hermite data, and the weights are obtained by the two given end curvatures. The conic precision property is based on the fact that the two conic Bezier curves are the same when the given G 2 Hermite data are sampled from a conic. Both the control points and weights of the resulting rational cubic Bezier curve are expressed in explicit form.

We prove that the Grothendieck standard conjecture of Lefschetz type on the algebraicity of operators and of Hodge theory holds for every smooth complex projective model of the fibre product , where is an elliptic surface over a smooth projective curve and is a family of K3 surfaces with semistable degenerations of rational type such that for a generic geometric fibre . We also show that holds for any smooth projective compactification of the Neron minimal model of an Abelian scheme of relative dimension over an affine curve provided that the generic scheme fibre is an absolutely simple Abelian variety with reductions of multiplicative type at all infinite places.

Let $X_0$ be a smooth hypersurface (not assumed generic) in projective space $\mathrm{P}^n$, $n \geq 3$ over the complex numbers, and $C_0$ a smooth rational curve on $X_0$. We are interested in the deformations of the pair $C_0 , X_0$. In this paper, we prove that if the first order deformations of the pair exist along certain first order deformations of the hypersurface $X_0$, then the twisted normal bundle $N_{C_0/ X_0}(1) = N_{C_0 / X_0} \otimes \mathcal{O}_{\mathcal{P}^n} (1) \vert {}_{C_0}$ is generated by global sections.

A large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties. For an elliptic curve over Q, the set of rational points forms a finitely generated abelian group. The ranks of these groups, when ranging over all elliptic curves, are conjectured to be evenly distributed between rank 0 and rank 1, with higher ranks being negligible. We will describe these conjectures and discuss some results on bounds for average rank, highlighting the recent work of Bhargava and Shankar.

Young's partition lattice L(m,n) consists of integer partitions having m parts where each part is at most n. Using methods from complex algebraic geometry, R. Stanley proved that this poset is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that it has a symmetric chain decomposition, which is a stronger property. Despite many efforts, this conjecture has only been proved for min(m,n)≤4. In this paper, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we find that the resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions.

We prove that the Chow quotient parameterizing configurations of n n points in P d \mathbb {P}^d which generically lie on a rational normal curve is isomorphic to M ¯ 0 , n \overline {\mathcal {M}}_{0,n} , generalizing the well-known d = 1 d=1 result of Kapranov. In particular, M ¯ 0 , n \overline {\mathcal {M}}_{0,n} admits birational morphisms to all the corresponding geometric invariant theory (GIT) quotients. For symmetric linearizations, the polarization on each GIT quotient pulls back to a divisor that spans the same extremal ray in the symmetric nef cone of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} as a conformal blocks line bundle. A symmetry in conformal blocks implies a duality of point-configurations that comes from Gale duality and generalizes a result of Goppa in algebraic coding theory. In a suitable sense, M ¯ 0 , 2 m \overline {\mathcal {M}}_{0,2m} is fixed pointwise by the Gale transform when d = m − 1 d=m-1 so stable curves correspond to self-associated configurations.

The goal of this paper is to demonstrate that all non-singular rational normal scrolls \(S(a_0,\ldots ,a_k)\subseteq \mathbb P ^N\), \(N =\sum _{i=0}^k(a_i)+k\), (unless \(\mathbb P ^{k+1}=S(0,\ldots ,0,1)\), the rational normal curve \(S(a)\) in \(\mathbb P ^a\), the quadric surface \(S(1,1)\) in \(\mathbb P ^3\) and the cubic scroll \(S(1,2)\) in \(\mathbb P ^4\)) support families of arbitrarily large rank and dimension of simple Ulrich (and hence indecomposable ACM) vector bundles. Therefore, they are all of wild representation type unless \(\mathbb P ^{k+1}\), \(S(a)\), \(S(1,1)\) and \(S(1,2)\) which are of finite representation type.

We calculate the Fourier-Deligne transform of the IC extension to ${\C}^{n+1}$ of the local system ${\mathcal L}_{\Lambda}$ on the cone over $\Conf_n({\P}^1)$ associated to a representation $\Lambda$ of $S_n$, where the length $n-k$ of the first row of the Young diagram of $\Lambda$ is at least $\frac{|\Lambda|-1}{2}$. The answer is the IC extension to the dual vector space ${\C}^{n+1}$ of the local system ${\mathcal R}_{\lambda}$ on the cone over the $k$-th secant variety of the rational normal curve in ${\P}^n$, where ${\mathcal R}_{\lambda}$ corresponds to the representation $\lambda$ of $S_k$, the Young diagram of which is obtained from the Young diagram of $\Lambda$ by deleting its first row. We also prove an analogous statement for $S_n$-local systems on fibers of the Abel-Jacobi map. We use our result on the Fourier-Deligne transform to rederive a part of a result of Michel Brion on Kronecker coefficients.

The problem of finding a non singular curve of genus gn without points of degree smaller than n was already studied in [1] when the genus is small (compared to the size of the finite field GF(q)). A construction of a family of curves without points of degree smaller than n was carried out by Clark and Elkies. The genus of such curves is bounded by gn large n. In this talk I give a ray class field construction of a family of curves over the finite field GF(q) without points of degree smaller than n and of genus smaller than q. References [1] Howe, E; Lauter, K; Top, J. Pointless curves of genus three and four. Seminaires et congres, 11 (2005), 125–141.

Given integers r>1, n>1 and q> n-2, we consider projective varieties X of dimension r+1 such that through n generic points of X passes a rational curve of degree q, contained in X. More precisely, we study the class X_{r+1,n}(q) of such varieties which moreover generate a projective space of the maximal dimension. We determine all varieties of a class X_{r+1,n}(q) when q is not equal to 2n-3. In particuliar, we show that there exists a variety X' in P^{r+n-1}, of minimal degree and a birational map F: X'---> X which sends a generic section of X' by a P^{n-1} onto a rational normal curve of degree q. Without hypothesis on q, we define a quasi-grassmannian structure on the space of the rational normal curves of degree q contained in a variety X of the class X_{r+1,n}(q). We prove that X is of the form described above if and only if this quasi-grassmannian structure is flat. We also give examples of varieties of the classes X_{r+1,3}(3) et X_{r+1,4}(5) which are not of this form.

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