I'm curious about a computation of Prop2.3 in The gonality conjecture on syzygies of algebraic curves of large degree by Ein and Lazarsfeld. Let $C$ be a smooth projective curve carrying a pencil $\alpha$ with degree $p+1$. Let $pr:C\times C_p\to C$ be the projection and $\sigma:C\times C_p\to C_{p+1}$ be defined by $(x,\xi)\mapsto x+\xi$, where all the subscripts refer to the symmetric product. Let $L_d$ be a divisor on $C$ of degree $d$ in the form $dx$ with $x\in C$ and $N_d=\det\sigma_*pr^*L_d$ a divisor on $C_{p+1}$. My question is why the degree of $N_d$ along $|\alpha|=\mathbb{P}^1$ is $d+$constant? Because we cannot distinguish invertible sheaves via fibers, I want to calculate $H^0(|\alpha|,i^*N_d)$, where $i:|\alpha|\hookrightarrow C_{p+1}$ is the embedding. Any help would be appreciated.


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