A \(g^r_d\) on a curve \(C\) is a linear system of degree \(d\) and projective dimension \(r\): a subspace \(V \subseteq H^0(\OO_C(D))\) with \(\deg D = d\) and \(\dim \PP V = r\). The gonality of \(C\) is the least \(d\) for which \(C\) carries a \(g^1_d\), equivalently the least degree of a nonconstant map \(C \to \PP^1\). The curve \(C\) is trigonal if its gonality is \(3\).
Gonality \(1\) means \(C \cong \PP^1\). For genus at least \(2\), gonality \(2\) means \(C\) is hyperelliptic; genus-\(1\) curves also have gonality \(2\). Every curve of genus \(g\) carries a \(g^1_d\) for \(d \geq \lfloor (g+3)/2 \rfloor\), and a general curve of genus \(g\) has no \(g^1_d\) below this bound.
A base-point-free \(g^r_d\) determines a morphism \(\phi\colon C\to\PP^r\) with \(\deg\phi^*\OO(1)=d\); for \(r=1\) this is a map \(C\to\PP^1\) of degree \(d\). The canonical system on a curve of genus \(g\ge2\) is a \(g^{g-1}_{2g-2}\).
For \(g=3\) and \(g=4\), \(\lfloor (g+3)/2\rfloor=3\), so every non-hyperelliptic curve of genus \(3\) or \(4\) is trigonal. A smooth plane quartic has a \(g^1_3\) for each of its points \(p\): the lines through \(p\) cut out \(p\) plus three further points.
Dependencies
T-MWDVLRiemann--Roch and Serre duality on a curveuses
Backlinks
D-CRVHYPHyperelliptic curves and their canonical mapsusesD-CRVBRILLNOETHERThe Brill--Noether number, and Brill--Noether general curvesuses