Relative canonical resolutions


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.

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