scholarly journals The Strong Maximal Rank conjecture and higher rank Brill–Noether theory

2021 ◽  
Author(s):  
Ethan Cotterill ◽  
Adrián Alonso Gonzalo ◽  
Naizhen Zhang
Keyword(s):  
Maximal Rank ◽  
Maximal Rank Conjecture ◽  
Rank Conjecture ◽  
Higher Rank

The maximal rank conjecture for non-special curves in ? n

1987 ◽  
Vol 196 (3) ◽  
pp. 355-367 ◽  
Author(s):  
Edoardo Ballico ◽  
Philippe Ellia
Keyword(s):  
Maximal Rank ◽  
Maximal Rank Conjecture ◽  
Rank Conjecture

The maximal rank conjecture for non-special curves in ?3

1985 ◽  
Vol 79 (3) ◽  
pp. 541-555 ◽  
Author(s):  
E. Ballico ◽  
Ph. Ellia
Keyword(s):  
Maximal Rank ◽  
Maximal Rank Conjecture ◽  
Rank Conjecture

scholarly journals Interpolation for Curves in Projective Space with Bounded Error

2019 ◽  
Author(s):  
Eric Larson
Keyword(s):  
Projective Space ◽  
Normal Bundle ◽  
Hyperplane Section ◽  
Maximal Rank ◽  
Special Linear ◽  
Maximal Rank Conjecture ◽  
Canonical Curve ◽  
Bounded Error ◽  
Rank Conjecture ◽  
Pass Through

Abstract 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].

The Chowla-Selberg Method for Fourier Expansion of Higher Rank Eisenstein Series

1985 ◽  
Vol 28 (3) ◽  
pp. 280-294 ◽  
Author(s):  
Audrey Terras
Keyword(s):  
Eisenstein Series ◽  
Fourier Expansion ◽  
Bessel Functions ◽  
Maximal Rank ◽  
Hecke Operators ◽  
Divisor Functions ◽  
Fourier Expansions ◽  
Higher Rank

AbstractThe terms of maximal rank in Fourier expansions of Eisenstein series for GL(n, ℤ) are obtained by an analogue of a method of Chowla and Selberg. The coefficients involve matrix analogues of divisor functions as well as K-Bessel functions for GL(n). The discussion involves a few properties of Hecke operators.

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