Talk:Canonical bundle

Canonical curves

There is more to say about canonical curves, a well-studied subject. In due course it will worth putting this on its own page, with a summary left here. Charles Matthews (talk) 15:41, 2 August 2009 (UTC)[reply]

The converse using Riemann-Roch needs the condition on the curve of linear normality, i.e. that the curve is embedded by a complete linear system. This is now explained briefly at homogeneous coordinate ring, but the concept really ought to be covered in greater depth somewhere else. In general there seems to be an absence on the site of the theory of algebraic space curves. Charles Matthews (talk) 08:18, 3 August 2009 (UTC)[reply]

Trying to check a key formula here. The space of quadratic differentials has dimension 3g - 3. The monomials of degree 2 in a basis of the differentials of the first kind number g(g + 1)/2. The difference works out at (g - 2)(g - 3)/2, which should therefore be the dimension of the quadrics through the curve. I think. Therefore if I'm reading Eisenbud's notation correctly, the index of the binomial coefficient is shifted one. Charles Matthews (talk) 09:32, 5 August 2009 (UTC)[reply]