Classification of two-dimensional algebraic projective semigroups

TL;DR

This paper classifies smooth projective algebraic surfaces over complex numbers that admit algebraic semigroup structures, providing a complete description for certain Kodaira dimensions and analyzing the moduli of such structures.

Contribution

It offers a full classification of surfaces with algebraic semigroup structures for Kodaira dimensions -∞, 0, and describes special elliptic surfaces for κ=1, also solving the moduli dimension problem for κ≥0.

Findings

Classified surfaces with semigroup structures for κ=-∞ and 0.

Described elliptic surfaces with semigroup laws for κ=1.

Solved the moduli dimension problem for surfaces with κ≥0.

Abstract

In this article, we address the classification of smooth projective algebraic surfaces over complex numbers admitting algebraic semigroup structures. We give a full description of those surfaces $S$, which has at least one non-trivial algebraic semigroup structure, when the Kodaira dimension of $S$ is $ -\infty$ and $ 0$. For the case "$ \kappa (S)=1$", we give a description of one special type of elliptic surfaces which admit non-trivial algebraic semigroup laws. \\ For a given surface $S$, it is an interesting problem to describe all algebraic semigroup structures on it and determine the dimension of this moduli. In this article, we solve this problem for case "$ \kappa (S)\ge 0$".