No local normalization

Question

It is well-known that the normalization of the complex curve $y^2-x^3-x^2$ has two maximal ideals. Can you explain me the generators of these ideals?

Answer 1

This is not exactly my meat, and others may do better at it.

Geometrically, the affine $t$-line curve maps onto your curve $y^2=x^3+x^2$ by $t\mapsto(t^2-1,t^3-t)$. You see that the two values $t=\pm1$ both map to the origin. These two points on the $t$-line give you your two maximal ideals in the normalized ring.

Algebraically, $t=y/x$ is in the fraction-field of $k[x,y]/(y^2-x^3-x^2)$, and is integral over it, because $t^2=x+1$. You see (same computation) that $k[x,y,t]/(y^2-x^3-x^2)=k[t]$, and that the maximal ideals $(t\pm1)$ both contract to $(x,y)\subset k[x,y]/(y^2-x^3-x^2)$.