Rational Normal Curve Research Articles

Rational conic fibrations containing special curves transverse to the fibers

Rational conic fibrations embedded in P N and containing an irreducible curve C transverse to the fibers are classified when C is either a plane curve or a rational normal curve of any degree d. This extends our previous results concerning lines, conics and twisted cubics. To emphasize the role of linear normality, the case in which C is a non-normal rational quartic is also discussed. Moreover, the rank of the d-th jet map of the surface at all points lying on such curves is evaluated.

Normal Bundles of Rational Normal Curves on Hypersurfaces

Normal Bundles of Rational Normal Curves on Hypersurfaces

A higgledy-piggledy set of planes based on the ABB-representation of linear sets

In this paper, we investigate the André/Bruck-Bose representation of certain Fq-linear sets contained in a line of PG(2,qt). We show that scatteredFq-linear sets of rank 3 in PG(1,q3) correspond to particular hyperbolic quadrics and that Fq-linear clubs in PG(1,qt) are linked to subspaces of a certain 2-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in PG(5,q).

Regularity and symbolic defect of points on rational normal curves

In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective 3-space. We give an explicit formula for the value of Castelnuovo–Mumford regularity for their ordinary powers. Moreover, we compare the m-th symbolic and ordinary powers for such ideals in order to show whenever the m-th symbolic defect is non-zero.

Twistors, Quartics, and del Pezzo Fibrations

It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the pluri-half-anti-canonical map from the twistor spaces. Specifically, each of the present twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result in Honda (“A new series of compact minitwistor spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil. Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among the present twistor spaces.

A characterisation of [formula omitted]-conics of PG(2,q3)

This article considers an Fq-conic contained in an Fq-subplane of PG(2,q3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). The main result characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to Fq-conics of PG(2,q3). The normal rational curves of interest are called 3-special, a property which describes how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. The proof uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q3) in PG(6,q), and the Bose representation of PG(2,q3) in PG(8,q).

An algebraic geometry perspective for the estimation of the directions of arrival

Directions of Arrival for Uniform Linear Arrays represent a widely studied topic in many fields of signal processing, with large attention on computational complexity, estimation accuracy and noise rejection. In this article we study the mathematical model behind Directions of Arrival from the point of view of algebraic geometry, focusing on its relation with the rational normal curve. On this basis, we give a novel interpretation for the root-MUSIC algorithm, that is a widely adopted estimation approach in signal processing. Furthermore, we propose some novel estimation techniques. The first one is based on the computation of the points on the rational normal curve that minimize the distance from the linear subspace defined by the measured Directions of Arrival. The others require the study of the secant varieties of the rational normal curve and the minimization of the distance between the point of the Grassmannian defined by the signal subspace and a certain secant variety. The results obtained from simulations in a noisy scenario show that our estimators are statistically consistent. One algorithm outperforms root-MUSIC over a wide range of scenarios, especially in presence of few snapshots and low Signal to Noise Ratio.

On linear systems with multiple points on a rational normal curve

We give a closed formula for the dimension of all linear systems in Pn with assigned multiplicity at arbitrary collections of points lying on a rational normal curve of degree n. In particular we give a purely geometric explanation of the speciality of these linear systems, which is due to the presence of certain subvarieties in the base locus: linear spans of points, secant varieties of the rational normal curve or joins between them.

Very free rational curves in Fano varieties

Let X be a projective variety and let C be a rational normal curve on X. We compute the normal bundle of C in a general complete intersection of hypersurfaces of sufficiently large degree in X. As a result, we establish the separable rational connectedness of a large class of varieties, including general Fano complete intersections of hypersurfaces of degree at least three in flag varieties, in arbitrary characteristic. In addition, we give a new way of computing the normal bundle of certain rational curves in products of varieties in terms of their restricted tangent bundles and normal bundles on each factor.

Rational Normal Curves and Hadamard Products

Given r>n general hyperplanes in mathbb P^n, a star configuration of points is the set of all the n-wise intersection of the hyperplanes. We introduce contact star configurations, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper, we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in mathbb P^2, we prove that the union of two contact star configurations has a special h-vector and, in some cases, this is a complete intersection.

ACM curves in multiprojective spaces

AbstractWe consider arithmetically Cohen–Macaulay (shortened to ACM) curves in a multiprojective space Y, i.e. curves $$T\subset Y$$ T ⊂ Y such that the restriction map $$H^0(L)\rightarrow H^0(L_{|T})$$ H 0 ( L ) → H 0 ( L | T ) is surjective for all line bundles L on Y such that $$H^0(L)\ne 0$$ H 0 ( L ) ≠ 0 . We prove that there are many such curves if all factors have dimension at least 3, but that T strongly resembles a rational normal curve if Y has $$\mathbb {P}^1$$ P 1 as one of its factors.

On Huisman’s conjectures about unramified real curves

AbstractLetX⊂ ℙnbe an unramified real curve withX(ℝ) ≠ 0. Ifn≥ 3 is odd, Huisman [9] conjectured thatXis anM-curve and that every branch ofX(ℝ) is a pseudo-line. Ifn≥ 4 is even, he conjectures thatXis a rational normal curve or a twisted form of such a curve. Recently, a family of unramifiedM-curves in ℙ3providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not evenM-curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.

Rank 3 quadratic generators of Veronese embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.

A Pascal's theorem for rational normal curves

Pascal's theorem gives a synthetic geometric condition for six points a , … , f in P 2 to lie on a conic. Namely, that the intersection points a b ¯ ∩ d e ¯ , a f ¯ ∩ d c ¯ , e f ¯ ∩ b c ¯ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for d + 4 points in 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 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.

Galois subspaces for the rational normal curve

We characterize all -dimensional linear subspaces of 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

A Topological View of Reed–Solomon Codes

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.

Reconstruction of rational ruled surfaces from their silhouettes

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 P3 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 P3 of these surfaces by exploiting the information contained in the singularities of the apparent contour.

Moment Identifiability of Homoscedastic Gaussian Mixtures

We consider the problem of identifying a mixture of Gaussian distributions with the same unknown covariance matrix by their sequence of moments up to certain order. Our approach rests on studying the moment varieties obtained by taking special secants to the Gaussian moment varieties, defined by their natural polynomial parametrization in terms of the model parameters. When the order of the moments is at most three, we prove an analogue of the Alexander–Hirschowitz theorem classifying all cases of homoscedastic Gaussian mixtures that produce defective moment varieties. As a consequence, identifiability is determined when the number of mixed distributions is smaller than the dimension of the space. In the two-component setting, we provide a closed form solution for parameter recovery based on moments up to order four, while in the one-dimensional case we interpret the rank estimation problem in terms of secant varieties of rational normal curves.

On varieties defined by large sets of quadrics and their application to error-correcting codes

Let U be a (k−12−1)-dimensional subspace of quadratic forms defined on Fk with the property that U does not contain any reducible quadratic form. Let V(U) be the points of PG(k−1,F) which are zeros of all quadratic forms in U. We will prove that if there is a group G which fixes U and no line of PG(k−1,F) and V(U) spans PG(k−1,F) then any hyperplane of PG(k−1,F) is incident with at most k points of V(U). If F is a finite field then the linear code generated by the matrix whose columns are the points of V(U) is a k-dimensional linear code of length |V(U)| and minimum distance at least |V(U)|−k. A linear code with these parameters is an MDS code or an almost MDS code. We will construct examples of such subspaces U and groups G, which include the normal rational curve, the elliptic curve, Glynn’s arc from Glynn (1986) and other examples found by computer search. We conjecture that the projection of V(U) from any k−4 points is contained in the intersection of two quadrics, the common zeros of two linearly independent quadratic forms. This would be a strengthening of a classical theorem of Fano, which itself is an extension of a theorem of Castelnuovo, for which we include a proof using only linear algebra.