On Zariski's theorem in positive characteristic

In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\dim(V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given ample linear system.Comment: 19 pages. Several typos have been fixed, and a couple of examples and pictures have been added. To appear in JEM

Natural Neutrino Dark Energy

A new class of neutrino dark energy models is presented. The new models are characterized by the lack of exotic particles or couplings that violate the standard model symmetry. It is shown that these models lead to several concrete predictions for the dark energy equation of state, as well as possible effects on the cosmic structure formation. These predictions, can be verified (or disproved) with future experiments. At this point, the strongest constraints on these models are obtained from big bang nucleosynthesis, and lead to new bounds on the mass of the lightest neutrino.Comment: 14 pages, 6 figure