Deformations and moduli of irregular canonical covers with $K^2=4p_g-8$

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $\varphi:X\to \mathbb{P}^N$, where $\varphi$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct families. We show that any deformation of $\varphi$ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $\varphi$ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that with the exception of one family, the deformations of $X$ are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g > 2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is a contrast with the moduli of surfaces with $K_X^2 = 2p_g - 4$, studied by Horikawa. There is a similarity and difference to the moduli of curves of genus $g\geq 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational.