Given two space curves defined by the equations 
$P\_1(x\_1,x\_2)=0,\dots,P\_{n-1}(x\_{n-1},x\_n)=0$ and $P\_1(x\_1,x\_2)=0,\dots,P\_{n}(x\_{n},x\_{n+1})=0$, respectively, is the genus of the former, say $g\_1$, smaller than or equal to the genus of the latter,
say $g\_2$? $P\_k$ are polynomials. For example, if $g\_2=0$, then the second curve admits a rational parametrization. Thus, also the first curve admits a rational parametrization and $g\_1=0$. Conversely, if $g\_1=0$, the second curve does not necessarily have a rational parametrization and, thus $g\_2\ge0$. Is the inequality $g\_1\le g\_2$ always true? 


More general question. Given a space curve in the variables $x\_1,\dots,x\_n$, let me define a second curve by adding a variable $x\_{n+1} $ and an additional polynomial constraint, is the genus of the second curve greater than or equal to the genus of the first one?



Genus of space curves given by the intersection of bivariate polynomial equations
Forgetting the variable $x\_{n+1}$ defines a non-constant, so dominant,
morphism from one curve to the other, so yes.


