Consider a plane curve $\mathcal{C}$ of degree $d$. We know that if a morphism $\varphi$ from $\mathcal{C}$ to a curve of genus $g\geq 2$ exists, then $\deg \varphi \leq (g'-1)/(g-1)$ where $g'$ is the genus of $\mathcal{C}$. As $g'\leq (d-1)(d-2)/2$, we deduce a bound for the degree of such morphism in function of $d$ $$\deg \varphi \leq \frac{(d-1)(d-2)-2}{2(g-1)}$$ However, this upper bound is not tight for $d\geq 4$. For $d=4,g=2$, this gives $\deg \varphi \leq 2$. However no such morphism exists for plane quartics. So noting the best bound $b(d,g)$, we have $b(4,2)=0,\; b(4,3)=1$. It seems to me that $b(d,g)$ should be at most linear in $d$, does there exist such results?
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