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The Castelnuovo-Severi says that given curves $X,Y,Z$ over a perfect field $k$ and nonconstant morphisms $f:X\rightarrow Y$ and $g:X \rightarrow Z$ which do not both factor through any morphism $X\rightarrow X'$, it holds that $$g(X)\leq \deg f\cdot g(Y) +\deg g \cdot g(Z)+(\deg f-1)(\deg g -1),$$ where $g(C)$ denotes the genus of the curve $C$.

What if there are 3 morphism from $X$ to other curves that satisfy the analogous property (of not factoring through some other morphism)? Can one get a better upper bound on the genus which is better than taking any 2 of the 3 morphisms and then applying the CS inequality to those 2?

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    $\begingroup$ Probably not, in general. If $X$ is the elliptic curve $x^2+y^2+(xy)^2=1$ then we get equality in the Castelnuovo inequality with the maps $x,y$ to $\mathbb{P}^1$ and the existence of a third map $xy$ won't help. $\endgroup$
    – Felipe Voloch
    Commented Jul 15, 2022 at 3:47

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