Every factor-compatible small cover over a product of polygons admits a smooth projective model as a finite quotient of a product of curves, and the graded mod 2 cohomology ring determines the Hodge diamond of that model.
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Small covers as pullbacks from the simplex are equivalently characterized by torsion-free odd-degree integral cohomology, vanishing of the first Steenrod square on even-degree mod 2 cohomology, and relations among integral and mod 2 Betti numbers.
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Symplectic and projective small covers over products of polygons
Every factor-compatible small cover over a product of polygons admits a smooth projective model as a finite quotient of a product of curves, and the graded mod 2 cohomology ring determines the Hodge diamond of that model.
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Small covers as pullbacks from the simplex
Small covers as pullbacks from the simplex are equivalently characterized by torsion-free odd-degree integral cohomology, vanishing of the first Steenrod square on even-degree mod 2 cohomology, and relations among integral and mod 2 Betti numbers.