Optimal and maximal singular curves - Archive ouverte HAL Access content directly
Journal Articles Contemporary mathematics Year : 2017

## Optimal and maximal singular curves

Yves Aubry
Annamaria Iezzi

#### Abstract

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F_q$. This bound enables us to provide explicit conditions on $q, g$ and $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N_q(g)+\pi-g$ rational points. Moreover, for $q$ a square, we study the set of pairs $(g,\pi)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi-g$ rational points.

#### Domains

Mathematics [math] Algebraic Geometry [math.AG]

### Dates and versions

hal-01212624 , version 1 (06-10-2015)

### Identifiers

• HAL Id : hal-01212624 , version 1
• ARXIV :
• DOI :

### Cite

Yves Aubry, Annamaria Iezzi. Optimal and maximal singular curves. Contemporary mathematics, 2017, Arithmetic, Geometry and Coding Theory, 686, pp.31--43. ⟨10.1090/conm/686/13776⟩. ⟨hal-01212624⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

154 View