New Examples of Calabi–yau 3-Folds and Genus Zero Surfaces

Abstract

We classify the subgroups of the automorphism group of the product of 4 projective lines admitting an invariant anticanonical smooth divisor on which the action is free. As a first application, we describe new examples of Calabi-Yau 3-folds with small Hodge numbers. In particular, the Picard number is 1 and the number of moduli is 5. Furthermore, the fundamental group is non-trivial. We also construct a new family of minimal surfaces of general type with geometric genus zero, K 2 = 3 and fundamental group of order 16. We show that this family dominates an irreducible component of dimension 4 of the moduli space of the surfaces of general type.

Calabi‐Yau Threefolds with Small Hodge Numbers

“…(-4, 6) (2, 4) Continued on next page 31 Some of these constructions are also discussed in [1,32,45]. 32 The cover manifold for these CICY quotients is a hypersurface in dP 6 ×dP 6 .…”

A twisted bicanonical system with base points

“…We first recall the construction of the 4−dimensional family M of minimal surfaces of general type with p g = q = 0, K 2 S = 3 and π 1 ≃ Z 4 ⋉Z 4 constructed in [BFNP14]. Consider the product X of 4 copies of P 1 .…”

“…Both authors are members of GNSAGA-INdAM. The second author is indebted with Jorge Neves for some inspiring discussions we had during the preparation of [BFNP14], that originated some of the results in section 3. Both authors thank the anonymous referee for correcting a mistake in Theorem 3.…”

“…Let M be the irreducible family of minimal surfaces of general type with p g = 0, K 2 = 3 and fundamental group Z 4 ⋉ Z 4 constructed in [BFNP14]. Then there is a 2−torsion η ∈ Pic(S) not divisible by 2 such that |2K + η| has exactly two simple base points.…”

“…In section 2 we describe the family M constructed in [BFNP14]. In section 3 we prove Theorem 1, which we split there in two distinct statements, namely propositions 5 and 6.…”