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No abstract uploaded!

It is shown that the punctual quotient scheme$Q$_{l}^{r}parametrizing all zero-dimensional quotients $$\mathcal{O}_{A^2 }^{ \oplus ^r } \to T$$ of length$l$and supported at some fixed point O∈$A$^{2}in the plane is irreducible.

In this paper, we compare some deterministic and probabilistic techniques in the study of upper bounds in problems related to certain mean square discrepancie with respect to balls in the$d$-dimensional unit torus, and show that the quality of these techniques depends in an intricate way on the dimension$d$under consideration.