Algebraic geometry look on space-filling curves Can space-filling curves be somehow described in terms of algebraic geometry? It appears to me that they shouldn't, but I'm not sure. Does anyone know of interesting papers on space-filling curves?

In fact there is an interesting discussion about this in Eisenbuds tome on Commutative Algebra (with a view towards AG). In particular, you may want to read his short and soft introduction to dimension theory chapter. He explains how the discovery of space filling curves helped algebraists understand the need for more precise notions of dimension than simply "number of free parameters."
(But Crostuls comment is correct, as you will see if you study dimension theory in, for example, Eisenbud.)
(Though possibly your question becomes more "interesting" if you work over a finite field, and you want to cover all of the $F_q$ points with a curve of some bounded degree.)
