Canonical curves of genus eight

Let C be a smooth complete algebraic curve of genus g and C2g−2 ⊂ Pg−1 the canonical model. It is generally difficult to describe its equations for higher genus. We restrict ourselves to the case of genus 8. If C has no g27, then C14 ⊂ P7 is a transversal linear section [G(2, 6) ⊂ P14] ∩H1 ∩ · · · ∩H7 of the 8-dimensional Grassmannian ([Muk2]). This is the case 〈8 〉 of the flowchart below. In this article we study the case where C has a g27 α. The system of defining equations of the canonical model is easily found from the following: Theorem (i) Assume that C has no g14. If α 2 ∼ = KC, then C is the in-tersection of the 6-dimensional weighted Grassmannian w-G(2, 5) ⊂ P(13: 26: 3) with a subspace P(13: 22), where w = (1, 1, 1, 3, 3)/2 (Case 〈7 〉 of Flowchart). Otherwise C is the complete intersection of three divisors of bidegree (1, 1), (1, 2) and (2, 1) in P2 × P2 (Case 〈6 〉 of Flowchart). (ii) Assume that C has a g14 but no g 2 6. Then C is the complete inter-section of four divisors of bidegree (1, 1), (1, 1), (0, 2) and (1, 2) in P1 × P4 (Case 〈5 〉 of Flowchart). Corollary A curve of genus 8 is a complete intersection of divisors in a variety X which is either a non-singular toric variety or a weighted Grass-mannian