We prove an analogue for $p$-adic coefficients of the Deligne--Laumon theorem on local acyclicity curves. That is, overconvergent $F$-isocrystal $E$ a relative curve $f:U\rightarrow S$ admitting good compactification, we show that cohomology sheaves $\mathbf{R}f_!E$ are isocrystals if and only has constant Swan conductor at infinity.

We consider the problem of determining the dimension of the space of bivariate splines C k( ), for all k. This problem is closely related to the question of whether C(
̂) is a free R-module. The main result is that C(
̂) is free if and only if j j has genus zero and C k( ) has the expected dimension for k = r + 1 (and hence for all k). We also obtain several interesting corollaries, including the...