Canonical curves of genus eight (Q1425414)

The authors study a description of all projective canonical curves \(C\) of genus \(8\), based on the existence of pencils and nets on \(C\) of small degree. It turns out that every such curve is a complete intersection of Cartier divisors in a variety \(X\) which is either a toric variety or a weighted Grassmannian. The most complicate case, to which the paper is mainly devoted, appears when \(C\) has no \(g^1_4\)'s but it has a net \(\alpha\) (of degree \(7\)) such that \(\alpha^2=K_C\). In this case, the authors prove that \(C\) is a complete intersection in a weighted Grassmannian \(wG(2,5)\subset \mathbb{P}(1^3:2^6:3)\).