TITLE: Steenrod operations in algebraic geometry
QUESTION [6 upvotes]: What are some applications of Steenrod operations (or similar constructions) in algebraic geometry? 
I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be happy if someone explained that in a comprehensible way. However, the primary motivation for asking the question is to see if such operations on the coherent (or de Rham) cohomology on algebraic varieties (in positive/mixed characteristic) have concrete geometric consequences.

REPLY [5 votes]: Not sure if this counts, but a much older application of $Sq^1$ (appearing as a Bockstein) and some of its cousins appears in Serre's paper on Witt vector cohomology. He constructed a family of operations, in characteristic $p$, $\beta_r: H^1(\mathcal{O}_X) \rightarrow H^2(\mathcal{O}_X)$ called the higher Bocksteins and it's reasonable to think of the first of these as a $Sq^1$.
A concrete application was given by Mumford who showed that a surface over a field of positive characteristic has smooth Picard scheme if and only if all of the Bocksteins on $H^1(\mathcal{O}_X)$ vanish. He later used this fact (amidst many others) in his and Bombieri's classification of algebraic surfaces in positive characteristic.