Rational Normal Curve Research Articles

Singularities of plane rational curves via projections

We consider the parameterization f=(f0:f1:f2) of a plane rational curve C of degree n, and we study the singularities of C via such parameterization. We use the projection from the rational normal curve Cn⊂Pn to C and its interplay with the secant varieties to Cn. In particular, we define via f certain 0-dimensional schemes Xk⊂Pk, 2≤k≤(n−1), which encode all information on the singularities of multiplicity ≥k of C (e.g. using X2 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 one to obtain information about the singularities from such schemes.

Regulating Hartshorne’s connectedness theorem

A classical theorem by Hartshorne states that the dual graph of any arithmetically Cohen–Macaulay projective scheme is connected. We give a quantitative version: If $$X \subset \mathbb {P}^n$$ is an arithmetically Gorenstein projective scheme of regularity $$r+1$$ , and if every irreducible component of X has regularity $$\le r'$$ , we show that the dual graph of X is $$\lfloor {\frac{r+r'-1}{r'}}\rfloor $$ -connected. The bound is sharp. We also provide a strong converse to Hartshorne’s result: every connected graph is dual to a suitable arithmetically Cohen–Macaulay projective curve of regularity $$\le $$ 3, whose components are all rational normal curves. The regularity bound is smallest possible in general. Further consequences are: (1) every graph is the Hochster–Huneke graph of a complete equidimensional local ring. (This answers a question by Sather–Wagstaff and Spiroff). (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.

Interpolation of Varieties of Minimal Degree

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.

Typical and admissible ranks over fields

Let \(X(\mathbb {R})\) be a geometrically connected variety defined over \(\mathbb {R}\) such that the set of all its complex points \(X(\mathbb {C})\) is non-degenerate. We introduce the notion of admissible rank of a point P with respect to X to be the minimal cardinality of a set \(S\subset X(\mathbb {C})\) of points such that S spans P and S is stable under conjugation. Any set evincing the admissible rank can be equipped with a 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 then there always exists a label for the generic element. We present two examples in which either the label doesn’t exist or the admissible rank is strictly bigger than the usual complex rank.

One-dimensional Schubert Problems with Respect to Osculating Flags

AbstractWe 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 zerodimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space allowing the points to collide. This gives rise to smooth covers (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 .Their(real) geometry turns out to be described by orbits of Schützenberger 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.

Irreducible components of Hilbert schemes of rational curves with given normal bundle

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.

Partition structure and the $A$-hypergeometric distribution associated with the rational normal curve

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.

Monodromy and K-theory of Schubert curves via generalized jeu de taquin

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 $S(\lambda_\bullet)$.

On the Effective Cone of Blown-up at n + 3 Points

ABSTRACTWe 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.

Projective curves of degree=codimension+2 II

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].

Factorization of point configurations, cyclic covers, and conformal blocks

We describe a relation between the invariants of n ordered points in projective d -space and of points contained in a union of two linear subspaces. 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 \overline 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. Consequently, 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.

Improved broadcast encryption schemes with enhanced security

A broadcast encryption scheme is one of important primitives to achieve message confidentiality in distributed network, which supports one-to-many encryption under insecure channels. In this paper, we show that any balanced incomplete block design constructed by perpendicular array can be used to realize secure BE schemes. Its broadcast rate, broadcast information rate and the upper bound of the number of collusion resistant are also obtained. According to the characteristics of the block, a more efficient broadcast encryption, using the strong part balanced incomplete block design constructed by rational normal curve is presented. This scheme is secure under an enhanced security model compared with the first construction. Particularly, even any two users are absent in the block, it can achieve fully collusion resistant. Moreover, an improved scheme is given based on the efficiency comparison of these two schemes, in which the broadcast rate is invariant while its broadcast information rate is significantly higher than that of the first construction. Finally, we propose a block reservation scheme, which provides the function of appending users dynamically.

Open problems in finite projective spaces

Apart from being an interesting and exciting area in combinatorics with beautiful results, finite projective spaces or Galois geometries have many applications to coding theory, algebraic geometry, design theory, graph theory, cryptology and group theory. As an example, the theory of linear maximum distance separable codes (MDS codes) is equivalent to the theory of arcs in PG(n,q); so all results of Section 4 can be expressed in terms of linear MDS codes. Finite projective geometry is essential for finite algebraic geometry, and finite algebraic curves are used to construct interesting classes of codes, the Goppa codes, now also known as algebraic geometry codes. Many interesting designs and graphs are constructed from finite Hermitian varieties, finite quadrics, finite Grassmannians and finite normal rational curves. Further, most of the objects studied in this paper have an interesting group; the classical groups and other finite simple groups appear in this way.

A structure theorem for 𝒮𝒰_{𝒞}(2) and the moduli of pointed rational curves

Let S U C ( 2 ) \mathcal {SU}_C(2) be the moduli space of rank 2 semistable vector bundles with trivial determinant on a smooth complex algebraic curve C C of genus g > 1 g>1 . We assume C C nonhyperellptic if g > 2 g>2 . In this paper we construct large families of pointed rational normal curves over certain linear sections of S U C ( 2 ) \mathcal {SU}_C(2) . This allows us to give an interpretation of these subvarieties of S U C ( 2 ) \mathcal {SU}_C(2) in terms of the moduli space of curves M 0 , 2 g \mathcal {M}_{0,2g} . In fact, there exists a natural linear map S U C ( 2 ) → P g \mathcal {SU}_C(2) \to \mathbb {P}^g with modular meaning, whose fibers are birational to M 0 , 2 g \mathcal {M}_{0,2g} , the moduli space of 2 g 2g -pointed genus zero curves. If g > 4 g>4 , these modular fibers are even isomorphic to the GIT compactification M 0 , 2 g G I T \mathcal {M}_{0,2g}^{GIT} . The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.

Schubert problems with respect to osculating flags of stable rational curves

At every point on the rational normal curve, there is an osculating flag. Consider Schubert problems with respect to r such flags, corresponding to r distinct points of the curve. It is natural to consider the r points as parametrized by the moduli space of r distinct points in P1 – the famous M0,r. One can then ask to extend the results on Schubert intersections to the compactification M0,r. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family G(d, n) over M0,r, whose fibers over M0,r are isomorphic to the Grassmannian G(d, n). For any Schubert problem, we construct a flat Cohen-Macaulay subfamily S of G(d, n) whose fibers over points of M0,r correspond to the solutions of that Schubert problem with respect to osculating flags. In the second part of the paper, we investigate the topology of the real points of S, in the case that the Schubert problem has finitely many solutions. We show that this space is a finite cover of M0,r(R). We give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux.

Geometry of the inversion in a finite field and partitions of PG(2k − 1, q) in normal rational curves

Let \({L = \mathbb{F}_{q^n}}\) be a finite field and let \({F=\mathbb{F}_q}\) be a subfield of L. Consider L as a vector space over F and the associated projective space that is isomorphic to PG(n − 1, q). The properties of the projective mapping induced by \({x \mapsto x^{-1}}\) have been studied in Csajbok (Finite Fields Appl. 19:55–66, 2013), Faina et al. (Eur. J. Comb. 23:31–35, 2002), Havlicek (Abh. Math. Sem. Univ. Hamburg 53:266–275, 1983), Herzer (Abh. Math. Sem. Univ. Hamburg 55:211–228 1985, Handbook of Incidence Geometry. Buildings and Foundations. Elsevier, Amsterdam, 1995). The image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer k, if q ≥ 2k − 1, then there are partitions of PG(2k − 1, q) in normal rational curves of degree 2k − 1. For smaller q the same construction gives partitions in (q + 1)-tuples of independent points.

Veronese quotient models of ̅M0,n and conformal blocks

The moduli space M0;n of Deligne-Mumford stable n-pointed rational curves admits morphisms to spaces recently constructed by Giansiracusa, Jensen, and Moon that we call Veronese quotients. We study divisors on M0;n associated to these maps and show that these divisors arise as rst Chern classes of vector bundles of conformal blocks. 0;n that receive morphisms from M0;n. From the perspective of Mori theory, this is tantamount to describing cer- tain semi-ample divisors on M0;n. This work is concerned with two recent constructions that each yield an abundance of such semi-ample divisors on M0;n, and the relationship between them. The rst comes from Geometric Invariant Theory (GIT), while the second from conformal eld theory. There are new natural birational models of M0;n obtained via GIT which are moduli spaces of pointed rational normal curves of a

A construction for infinite families of semisymmetric graphs revealing their full automorphism group

We give a general construction leading to different non-isomorphic families $\varGamma_{n,q}(\mathcal{K})$ of connected q-regular semisymmetric graphs of order 2q n+1 embedded in $\operatorname{PG}(n+1,q)$ , for a prime power q=p h , using the linear representation of a particular point set $\mathcal{K}$ of size q contained in a hyperplane of $\operatorname{PG}(n+1,q)$ . We show that, when $\mathcal{K}$ is a normal rational curve with one point removed, the graphs $\varGamma_{n,q}(\mathcal{K})$ are isomorphic to the graphs constructed for q=p h in Lazebnik and Viglione (J. Graph Theory 41, 249---258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897---902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q?n+3 or q=p=n+2, n?2, we obtain their full automorphism group from our construction by showing that, for an arc $\mathcal{K}$ , every automorphism of $\varGamma_{n,q}(\mathcal{K})$ is induced by a collineation of the ambient space $\operatorname{PG}(n+1,q)$ . We also give some other examples of semisymmetric graphs $\varGamma _{n,q}(\mathcal{K})$ for which not every automorphism is induced by a collineation of their ambient space.

Tropical decomposition of Young’s partition lattice

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.

Birkhoff Strata of Sato Grassmannian and Algebraic Curves

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum ΣS contains a subset Wŝ of points for which each fiber of the corresponding tautological subbundle TBWS is closed with respect to multiplication. Algebraically TBWS is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle TBW∅ represents the tower of families of normal rational (Veronese) curves of all degrees. For W1 such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles TBW1,2,..,n represent families of plane (n + 1, n + 2) curves (trigonal curves at n = 2) and space curves of genus n. Two methods of regularization of singular curves contained in TBWŝ, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.