  Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We
show that the inclusion of the space $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ of
real parts of nonflat proper algebraic null immersions $M\to\mathbb{C}^n$,
$n\ge 3$, into the space $\mathrm{CMI}_*(M,\mathbb{R}^n)$ of complete nonflat
conformal minimal immersions $M\to\mathbb{R}^n$ of finite total curvature is a
weak homotopy equivalence. We also show that the $(1,0)$-differential
$\partial$, mapping $\mathrm{CMI}_*(M,\mathbb{R}^n)$ or $\Re
\mathrm{NC}_*(M,\mathbb{C}^n)$ to the space $\mathscr{A}^1(M,\mathbf{A})$ of
algebraic $1$-forms on $M$ with values in the punctured null quadric
$\mathbf{A} \subset \mathbb{C}^n\setminus\{0\}$, is a weak homotopy
equivalence. Analogous results are obtained for proper algebraic immersions
$M\to\mathbb{C}^n$, $n\ge 2$, directed by a flexible or algebraically elliptic
punctured cone in $\mathbb{C}^n\setminus\{0\}$.
