By improving the vanishing theorem of Hirschowitz, we prove an existence theorem of the reduced irreducible plane curve with ordinary singularities at given points in general position, which improves an earlier result by Greuel, Lossen and Shustin.

A non-tetragonal curve of genus 8 is a complete intersection of divisors in either $\mathbf{P}^2 \times \mathbf{P}^2$, a 6-dimensional weighted Grassmannian or the 8-dimensional Grassmannian.

Pour-El & Richards [3] discussed an ad hoc computability structure in an effectively separable Hilbert space taking as an effective generating set a slightly modified one from the original orthonormal basis. We show that an application of the Poincaré-Wigner orthogonalizing procedure to Pour-El & Richards' modified system gives an orthonormal effective generating set which yields a third computability structure.