Normalization resolves the singularities of a projective curve

Regard the projective curve $C$ as an integral closed subscheme of ${\mathbf{P}}_k^r$. Its nonempty standard affine charts are $C_i=\mathrm{Spec}{A}_i$, where the $A_i$ are finite-type domains with common function field $K=k(C)$. Put $B_i$ equal to the integral closure of $A_i$ in $K$. By [F1], $B_i$ is finite over $A_i$. On $C_i\cap C_j=D(x_j\mathrm{/}{x}_i)$ these closures localize to the same subring of $K$. Thus their affine spectra glue by [F2] to a scheme $C^{\nu }$ with a finite morphism $\nu :C^{\nu }\to C$. Each $B_i$ is a finite algebra over the finite-type $k$-algebra $A_i$, hence finite type over $k$; the finite chart cover makes $C^{\nu }$ a finite-type $k$-scheme. This is the normalization: its affine rings and their localizations are exactly the integral closures in $K$.

Each $B_i$ is a normal domain with fraction field $K$. Lying over makes $\nu$ surjective. Clearing the denominators in $K$ of a finite set of $A_i$-module generators of $B_i$ gives a nonzero $a_i\in A_i$ with $(B_i{)}_{a_i}=(A_i{)}_{a_i}$, so $\nu$ is birational. Moreover $\dim B_i={\mathrm{trdeg}}_{k}K=1$ by [F2]; hence $C^{\nu }$ is an integral normal curve. By [F4] it is projective, and by [F3] the perfectness of $k$ makes it nonsingular.

The scheme $C^{\nu }$ therefore supplies the required nonsingular projective model with its finite birational normalization map. The construction uses affine integral closure over the actual field $k$, so it applies to every perfect field, not only the algebraically closed classical case.