Construction of surfaces of general type from elliptic surfaces via $${\mathbb{Q}}$$ -Gorenstein smoothing

Abstract

We present methods to construct interesting surfaces of general type via 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 pg = 0, π1 = Z/2Z, and K 2 ≤ 4.

A complex surface of general type with 𝑝_{𝑔}=0, 𝐾²=2 and 𝐻₁=ℤ/4ℤ

“…In Section 4 we show that the obstruction space of a global Q-Gorenstein smoothing of the singular surface X is zero by a similar strategy as in J. Keum-Y. Lee-H. Park [9]. Hence these local smoothings can be glued to a global Q-Gorenstein smoothing of the whole singular surface X.…”

A fake projective plane constructed from an elliptic surface with multiplicities (2, 4)

“…Recall that if an elliptic surface with p g = q = 0 has a μI 9 -fibre, then its Jacobian surface is a rational elliptic surface with a section having one I 9 -fibre, hence has three I 1 -fibres (cf. [8,Lemma 3.3]). Thus the original surface has three I 1 -fibres, multiple or non-multiple.…”

Godeaux, Campedelli, and surfaces of general type with χ=4 and 2≤K2≤8

“…In 2007, Lee and Park introduced a method for constructing algebraic surfaces using ‐Gorenstein smoothing. Lee and Park used this technique to construct many families of surfaces which include surfaces of general type with , , and , , . Later on H. Park, J.…”