The minimal degree of plane models of algebraic curves and double coverings (Q845796)

Fix a large genus \(g\), say \(g \geq 12\). For any smooth curve \(C\) let \(S(C,2)\) be the minimal degree of a (possibly singular) plane model of \(C\), i.e. the minimal integer \(d\) such that \(C\) has a \(g^2_d\) not composed with an involution. This classical invariant was considered in [\textit{C. Keem} and \textit{G. Martens}, Math. Nachr. 281, No. 12, 1791--1798 (2008; Zbl 1162.14020)], who proved that \(C\) is bi-elliptic (resp. hyperelliptic) if and only if \(s(C,2) = g+1\) (resp. \(s(C,2)=g+2\)). Here the authors goes much further. They prove that \(s(C,2)\neq g\). Also, further results are proved here. Assume \(s(C,2) = g+2-t\) with \(t\geq 0\); if \(g \geq (t+1)(t+2)\), then \(C\) is a double covering of a curve of genus \(g' \leq t(t+1)\) and \(g \geq 2(g'+t+1)\); if \(g \geq 3t^2+4t-1\), then \(g' \leq t\). If \(C\) is a double covering of a hyperelliptic curve of genus \(g'\), then \(s(C,2) \geq g-2g'+3\); if either \(g\) is even and \(g \geq 4g'\) or \(g\) is odd and \(g \geq 6g'\), then \(s(C,2) = g-2g'+3\) and \(C\) has a plane model with one singular point of multiplicity \(g-2g'-1\) and ordinary nodes as its other singularities. They also give more detailed informations for low \(g\) (e.g. if \(g \geq 8\), \(g\neq 9,11\) and \(C\) is a double covering of a curve of genus \(2\), then \(s(C,2)=g-1\)).