Let $H_0$ be a discrete periodic  Schr\"odinger operator on $\Z^d$:

$$H_0=-\Delta+V,$$ where $\Delta$ is the discrete Laplacian and $V:\Z^d\to \R$ is periodic.    We prove that  for any $d\geq3$,    the Fermi variety at every energy level  is irreducible  (modulo periodicity).  For $d=2$,    we prove that the Fermi variety at every energy level except for the average of  the potential    is irreducible  (modulo periodicity) and  the Fermi variety at the average of  the potential has at most two irreducible components  (modulo periodicity). 

This is sharp since for  $d=2$ and a constant potential  $V$,   

the Fermi variety at  $V$-level  has exactly  two irreducible components (modulo periodicity).  

In particular,  we show that  the Bloch variety  is irreducible 

(modulo periodicity)  for any $d\geq 2$.