Interpolation for Curves in Projective Space with Bounded Error

Given $n$ general points $p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$ it is natural to ask whether there is a curve of given degree $d$ and genus $g$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if \begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\end{equation*}The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a “bounded-error analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality \begin{equation*}n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\end{equation*}Note that the $-3$ cannot be replaced with $-2$ without introducing exceptions (as a canonical curve in ${\mathbb{P}}^3$ can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle $N_C(-1)$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least \begin{equation*}\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\end{equation*} As explained in [7], these results play a key role in the author’s proof of the maximal rank conjecture [9].

ON GEODESICS IN SUBRIEMANNIAN GEOMETRY II

We study a subRiemannian geometry induced by 2 specific vector fields in ℝ3, and obtain the canonical curve whose tangents provide the missing direction.

CANONICAL CURVES IN ℙ3

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

On the number of moduli of extendable canonical curves

Let C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12.