On Huisman’s conjectures about unramified real curves

Let X ⊂ ℙ n be an unramified real curve with X(ℝ) ≠ 0. If n ≥ 3 is odd, Huisman [9] conjectured that X is an M-curve and that every branch of X(ℝ) is a pseudo-line. If n ≥ 4 is even, he conjectures that X is a rational normal curve or a twisted form of such a curve. Recently, a family of unramified M-curves in ℙ3 providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not even M-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.

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.

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

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, 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 ω 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 RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω 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. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

One-dimensional Schubert Problems with Respect to Osculating Flags

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 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.

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].

MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

Let C ⊂ ℙn+1 be a rational normal curve and let X ⊂ ℙn be one of its tangential projection. We describe the X-rank of a point P ∈ ℙn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.