The minimal degree of plane models of algebraic curves and double coverings

Abstract

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g = 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.

The lattice size of a lattice polygon

“…If g ≥ 6 then s 2 (C) = g + 1 if and only if C is bi-elliptic. See [18] and the references therein for proofs.…”

“…The invariant s 2 (C) has seen study in the past [12,18,20] but is not wellunderstood. On the other hand we are unaware of existing literature explicitly devoted to s 1,1 (C), even though for hyperelliptic curves the notion did make an appearance [15] in the context of cryptography.…”

“…If g ≥ 6 then s 2 (C) = g + 1 if and only if C is bi-elliptic. See [18] and the references therein for proofs. In the case of s 1,1 (C) we prove an analogous statement:…”