The Noether inequality for 3‐folds and three moduli spaces with minimal volumes

Abstract We establish the Noether inequality for all projective 3‐folds of general type with geometric genus where is the canonical volume. This result resolves all remaining cases of the Noether inequality for 3‐folds. We further investigate the moduli spaces of canonical 3‐folds with small genera and minimal volumes. For a 3‐fold of general type with geometric genus 2 and with minimal canonical volume , we prove that its canonical model is a hypersurface of degree 16 in , which gives an explicit description of its canonical ring. This implies that the coarse moduli space , parameterizing all canonical 3‐folds with canonical volume and geometric genus 2, is an irreducible unirational variety of dimension 189. Parallel studies show that is irreducible, unirational, and 236‐dimensional, and that is irreducible, unirational, and 270‐dimensional. As being conceived, every member in these three families is simply connected.