TITLE: Is there an equivalent concept of a "variety" for SAT?
QUESTION [5 upvotes]: Couldn't find anything via google - I was wondering what work is out there looking at SAT problems from the perspective akin to an algebraic variety, e.g. a set of variables $X_1=$true, $X_2=$false, .., $X_k=$true, etc define a set of propositions which are all satisfed by that particular set of values in much the same way that a variety defines a set of polynomials that all happen to vanish at the same points.
Thanks.

REPLY [2 votes]: For one variety over one finite field, the algebraic geometry translation of a SAT problem is not all that useful.  Switching between the two descriptions is computationally trivial. Algebraic geometry doesn't help with finding solutions so much as describing relations between solutions. SAT problems do not lead to structured varieties like elliptic curves or algebraic surfaces where algebraic geometry comes into its own.
The advantage of geometry is when there are many varieties, or many fields and rings in which to solve the equations, or both. 
This is roughly the idea of Aaronson and Wigderson's paper on Algebraization as a barrier to $P \neq NP$ proofs.  General arguments about computation with bits, thought of as facts about polynomials or varieties where we look for solutions mod $2$, also would work over finite fields with more than $2$ elements, or more general rings than that. In those more general settings they can arrange oracles for which $P = NP$, so things are not that easy.
Geometric Complexity Theory proposed by Mulmuley is an application of algebraic geometry to complexity and an approach to the $P=NP$ problem, but the geometry does not come from mapping SAT problems to algebraic equations mod $2$.  The idea is to look for algebraic problems in invariant theory and group representation theory that are close enough analogues of $P=NP$ that solving the algebra problems would have implications for computational complexity theory.  Algebraic geometry appears in GCT because it is a tool for the algebra problems that appear.