This paper presents new examples of projective surfaces of general type over $\mathbb{C}$ with canonical map of degree $ 3 $ onto a surface of general type. Very few examples are known of such surfaces and some of the examples in this paper present the new feature of having the canonical map not a morphism (i.e. the canonical linear system with base points).

New examples of canonical covers of degree 3

For a convex lattice polytope having at least one interior lattice point, a lower bound for its volume is derived from Hibi’s lower bound theorem for the $$h^{*}$$
 -vector. On the other hand, it is known that the sectional genus of a polarized variety has an upper bound, which is an extension of the Castelnuovo bound for the genus of a projective curve. In this paper, we prove the equivalence of these two bounds. Namely, a polarized toric variety has maximal sectional genus if and only if its associated polytope has minimal volume. This is a generalization of the known fact that polytopes corresponding to the anticanonical bundles of Gorenstein toric Fano varieties are reflexive polytopes (whose typical examples are minimal volume polytopes with only one interior lattice point).

Sectional genus and the volume of a lattice polytope

. It is known that the optimal Noether inequality vol( X ) ≥ 43 p g ( X ) − 103 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.

Algebraic threefolds of general type with small volume

We prove the optimal Noether-Severi inequality that $\mathrm{vol}(X) \ge \frac{4}{3} \chi(\omega_{X})$ for all smooth and irregular $3$-folds $X$ of general type. This answers an open question of Z. Jiang in dimension three. Furthermore, for those $X$ attaining the equality, we completely describe their canonical models and show that the topological fundamental group $\pi_1(X) \simeq \mathbb{Z}^2$. As a corollary, we obtain for the same $X$ another optimal inequality that $\mathrm{vol}(X) \ge \frac{4}{3}h^0_a(X, K_X)$ where $h^0_a(X, K_X)$ stands for the continuous rank of $K_X$, and we show that $X$ attains this equality if and only if $\mathrm{vol}(X) = \frac{4}{3}\chi(\omega_{X})$.

Noether-Severi inequality and equality for irregular threefolds of general type

In this note, we construct nine families of projective complex minimal surfaces of general type having the canonical map of degree 8 and irregularity 0 or 1. For six of these families the canonical system has a non trivial fixed part.

Some unlimited families of minimal surfaces of general type with the canonical map of degree 8.

In this paper, we study minimal algebraic surfaces of general type with K2=3pg−5. Since the case pg≤4 has been studied by Horikawa, Zucconi, and Bauer, we always assume pg≥5 here. We prove that for...

Songbo Ling

Papers

Algebraic surfaces of general type with