Algebraic threefolds of general type with small volume

2022/04/05 by Yong Hu, Tong Zhang, Hu, Yong +1 · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2204.02222

Abstract

It is known that the optimal Noether inequality vol(X) ≥ (4)/(3)p_g(X) - (10)/(3) holds for every 3-fold X of general type with p_g(X) ≥ 11. In this paper, we give a complete classification of 3-folds X of general type with p_g(X) ≥ 11 satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when p_g(X) ≥ 23. We also establish the second and third optimal Noether inequalities for 3-folds X of general type with p_g(X) ≥ 11. These results answer two open questions raised by J. Chen, M. Chen and C. Jiang, and in dimension three an open question raised by J. Chen and C. Lai. A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of p_g(X) modulo 3.

Cited by

• On the moduli space of stable surfaces with p_g=1 realizing the minimal volume
• On Automorphism Groups of (1,2)-Surfaces

Related

• The Noether inequality for threefolds and three moduli spaces with minimal volumes
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