# Degree of a divisor along a subscheme

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.