Abstract Let $${\mathcal {F}} \, = \, (\dots {\mathop {\rightarrow }\limits ^{\partial _{n+1}}} {\mathcal {F}}_n {\mathop {\rightarrow }\limits ^{\partial _n}} {\mathcal {F}}_{n-1}{\mathop {\rightarrow }\limits ^{\partial _{n-1}}} \dots \dots {\mathop {\rightarrow }\limits ^{\partial _1}} {\mathcal {F}}_0 \rightarrow {\mathfrak {R}} \rightarrow 0)$$ F = ( ⋯ → ∂ n + 1 F n → ∂ n F n - 1 → ∂ n - 1 ⋯ ⋯ → ∂ 1 F 0 → R → 0 ) be a free resolution over the group ring $${\mathfrak {R}}[\Phi ]$$ R [ Φ ] where $${\mathfrak {R}}$$ R is commutative and $$\Phi $$ Φ is finite. The $$n^{th}$$ n th syzygy $$\Omega _n^{{\mathfrak {R}}[\Phi ]}$$ Ω n R [ Φ ] is the stable class of $$\textrm{Im}(\partial _n)$$ Im ( ∂ n ) and has a tree structure with roots which do not extend infinitely downwards. We show that $$\Omega _3^{{\mathfrak {R}}[Q_{8p}]}$$ Ω 3 R [ Q 8 p ] has infinitely many isomorphically distinct modules at the minimal level when $$\,{\mathfrak {R}} = {\mathbb {Z}}[C_\infty ]$$ R = Z [ C ∞ ] is the integral group ring of the infinite cyclic group and $$Q_{8p}$$ Q 8 p is the quaternion group of order 8 p where $$p \ge 3$$ p ≥ 3 is prime. This poses severe difficulties in attempting to solve the D (2) problem of CTC Wall for the groups $$C_\infty \times Q_{8p}$$ C ∞ × Q 8 p

Abstract We introduce the notion of a combinatorial K3 surface. Those form a certain class of type III semistable K3 surfaces and are completely determined by combinatorial data called curve structures. Emphasis is put on degree 2 combinatorial K3 surfaces, but the approach can be used to study higher degree as well. We describe elementary modifications both in terms of the curve structures as well as on the Picard groups. Together with a description of the nef cone in terms of curve structures, this provides an approach to explicitly computing the Mori fan of the Dolgachev–Nikulin–Voisin family in degree 2.

Abstract We investigate the “natural” locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C – not necessarily smooth – the Abel-Jacobi map from the smooth locus $$C^{\textrm{sm}}$$ C sm into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing $$C^{\textrm{sm}}$$ C sm except under certain particular circumstances which we describe explicitly. As a consequence, we deduce that the Abel-Jacobi map has closed image except in certain explicitly described circumstances, and that it is always a closed embedding for irreducible curves not isomorphic to $$\textbf{P}^1$$ P 1 .