A bound on the number of points of a curve in a projective space over a finite field (Q2869131)

In a series of papers, Homma and Kim proved that for any plane curve \(C/\mathbb{F}_q\) of degree \(d\) without \(\mathbb{F}_q\)-linear components, the number \(N_q(C)\) of \(\mathbb{F}_q\)-points of \(C\) is at most \((d-1)q+1\), with one single exception: the curve over \(\mathbb{F}_4\) with equation \((x+y+z)^4+(xy+yz+zx)^2+xyz(x+y+z)=0\) has \(14\) rational points.NEWLINENEWLINEIn this paper, the author generalizes this bound to curves in the projective space \(\mathbb{P}^r\) for \(r\geq 3\). The main result states that an absolutely irreducible curve over \(\mathbb{F}_q\) of degree \(d\) in \(\mathbb{P}^r\), which is not contained in any plane, satisfies \(N_q(C)\leq (d-1)q+1\). The main ingredient in the proof is the order-sequence of the curve as in Stöhr-Voloch theory.NEWLINENEWLINEFor the entire collection see [Zbl 1253.00023].