# Relative canonical resolutions

## Background

We briefly recall the setting: Let $$C\subset \mathbb{P}^{g-1}$$ be a canonically embedded curve of genus $$g$$ admitting a complete base point free a pencil of divisors $$|D|=g^1_k$$ of degree $$k$$. If we denote by $$\overline{D_\lambda}\subset \mathbb{P}^{g-1}$$ the linear span of the divisor, then $$X=\bigcup_{\lambda\in \mathbb{P}^1}\overline{D_\lambda}\subset \mathbb{P}^{g-1}$$ is a $$(k-1)$$-dimensional rational normal scroll of degree $$f=g-k+1$$. On the other hand this scroll is the image of $$j:\mathbb{P}(\mathscr{E})\to \mathbb{P} H^0(\mathbb{P}(\mathscr{E}),\mathscr{O}_{\mathbb{P}(\mathscr{E})}(1))= \mathbb{P}^{g-1}$$ where $$\mathscr{E}=\mathscr{O}_{\mathbb{P}^1}(e_1)\oplus\dots\oplus\mathscr{O}_{\mathbb{P}^1}(e_d)$$ and $$f=e_1+\dots+e_d$$. It is furthermore known, that the Picard group $$\text{Pic}(\mathbb{P}(\mathscr{E}))$$ of the projective bundle $$\pi:\mathbb{P}(\mathscr{E})\to \mathbb{P}^1$$ is generated by the ruling $$R=[\pi^*\mathscr{O}_{\mathbb{P}^1}(1)]$$ and the hyperplane class $$H=[j^*\mathscr{O}_{\mathbb{P}^{g-1}}(1)]$$ with intersection products $$H^d=f, \ \ H^{d-1}\cdot R=1, \ \ R^2=0.$$ Hence, we will write a line bundle $$\mathscr{O}_{\mathbb{P}(\mathscr{E})}(aH+bR)$$ in the following form $$\mathscr{O}_{\mathbb{P}(\mathscr{E})}(aH+bR) = \pi^*(\mathscr{O}_{\mathbb{P}^1}(b))(aH).$$ The relative canonical resolution of $$C$$ with respect to the pencil $$g^1_k$$ is the minimal free resolution of $$C\subset \mathbb{P}(\mathscr{E})$$ in terms of generators of the scroll. By results of F.-O. Schreyer this resolution has the form $$F_\bullet:= \ 0 \to \pi^*N_{k-2}(-k) \to \pi^*N_{k-3}(-k+2) \to \dots \to \pi^*N_1(-2) \to \mathscr{O}_{\mathbb{P}(\mathscr{E})}\to \mathscr{O}_C \to 0$$ with $$N_i=\sum_{j=1}^{\beta_i}\mathscr{O}_{\mathbb{P}^1}(a_j^{(i)})$$ and $$\beta_i=\frac{i(k-2-i)}{k-1}\binom{k}{i+1}$$. The resolution is furthermore self-dual, i.e., $$\mathscr{H}om(F_\bullet, \mathscr{O}_{\mathbb{P}(\mathscr{E})}(-kH+(f-2)R))\cong F_\bullet$$.

Since the $$N_i$$'s appearing in the relative canonical resolution are bundles on $$\mathbb{P}^1$$, they split into a direct sum of line bundles. We call a bundle $$N=\oplus_{j=1}^{\beta_i}\mathscr{O}_{\mathbb{P}^1}(a_j)$$ on $$\mathbb{P}^1$$ balanced if $$\max |a_i-a_j|\leq 1$$. It is an interesting problem, whether the bundles $$N_i$$ in the relative canonical resolution of a general canonical curve of genus $$g$$ together with a general pencil of degree $$k$$ are balanced. Knowing the answer to this problem has various applications in the study of the corresponding Hurwitz spaces $$\mathscr{H}_{g,k}$$. The data displayed on this webpage are the Betti numbers of relative canonical resolutions for various cases.

© AG Schreyer