# The maximal rank conjecture

@article{Larson2018TheMR, title={The maximal rank conjecture}, author={Eric Larson}, journal={arXiv: Algebraic Geometry}, year={2018} }

Let C be a general curve of genus g, embedded in P^r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C.

## 19 Citations

Degenerations of Curves in Projective Space and the Maximal Rank Conjecture

- Mathematics
- 2018

In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank…

On the strong maximal rank conjecture in genus 22 and 23

- Mathematics
- 2018

We develop new methods to study tropicalizations of linear series and show linear independence of sections. Using these methods, we prove two new cases of the strong maximal rank conjecture for…

Effective divisors on Hurwitz spaces

- Mathematics
- 2018

We prove the effectiveness of the canonical bundle of several Hurwitz spaces of degree k covers of the projective line from curves of genus 13

The strong maximal rank conjecture and moduli spaces of curves

- Mathematics
- 2018

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This…

Limit linear series and ranks of multiplication maps

- MathematicsTransactions of the American Mathematical Society
- 2020

We develop a new technique to study ranks of multiplication maps for linear series via limit linear series and degenerations to chains of elliptic curves. We prove an elementary criterion and apply…

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

- MathematicsJournal of the London Mathematical Society
- 2021

In this paper, we compute the cohomology class of certain ‘special maximal‐rank loci’ originally defined by Aprodu and Farkas. By showing that such classes are non‐zero, we are able to verify the…

The generality of a section of a curve

- MathematicsJournal of the London Mathematical Society
- 2016

This paper considers the following fundamental problem about intersections in projective space: When is the intersection of a (varying) curve with a (fixed) hypersurface a general set of points on…

Effectivity of Farkas classes and the Kodaira dimensions of M_{22} and M_{23}.

- Mathematics
- 2018

We develop new methods to study tropicalizations of linear series and show linear independence of algebraic sections. Using these methods, we prove two outstanding cases of the strong maximal rank…

Recent Developments in Brill-Noether Theory

- Mathematics
- 2021

We briefly survey recent results related to linear series on curves that are general in various moduli spaces, highlighting the interplay between algebraic geometry on a general curve and the…

Interpolation for Brill–Noether curves in
$${\mathbb {P}}^4$$
P
4

- MathematicsEuropean Journal of Mathematics
- 2020

We compute the number of general points through which a general Brill–Noether curve in $${\mathbb {P}}^4$$ P 4 passes. We also prove an analogous theorem when some points are constrained to lie in a…

## References

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Degenerations of Curves in Projective Space and the Maximal Rank Conjecture

- Mathematics
- 2018

In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank…

Tropical independence II: The maximal rank conjecture for quadrics

- Mathematics
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Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a…

The generality of a section of a curve

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- 2016

This paper considers the following fundamental problem about intersections in projective space: When is the intersection of a (varying) curve with a (fixed) hypersurface a general set of points on…

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Let X be a complete nonsingular curve of genus g defined over an algebraically closed field Ic of any characteristic, let J be the jacobian variety. Denote by G;r the set of points of J which are…

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Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f :…

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0. Introduction (a) Statement of the problem and of the main theorem; some references 233 (b) Corollaries of the main theorem 236 (c) Role of the Brill-Noether matrix 238 (d) Heuristic reasoning for…

Embeddings of general curves in projective spaces: the range of the quadrics

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Let $$ C \subset {\mathbb{P}^r} $$ be a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either $$ {h^0}\left(…

Interpolation for Curves in Projective Space with Bounded Error

- MathematicsInternational Mathematics Research Notices
- 2019

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…