Construction of surfaces of general type from elliptic surfaces via Q-Gorenstein smoothing

We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new examples of a minimal surface of general type with $p_g=0$, $\pi_1=\mathbb{Z}/2\mathbb{Z}$, and $K^2\le 4$.