Catanese's paracanonical system conjecture for surfaces

Let $S$ be a smooth projective surface over $\mathbb{C}$ without irrational pencils. The paracanonical system is denoted by $\{K_S\}$, and $p_g(S)$ denotes the geometric genus.

Catanese's conjecture. The dimension of the paracanonical system satisfies

$$\dim\{K_S\}\leq p_g(S).$$

Green and Lazarsfeld gave a positive answer to this conjecture in 1987, so the conjecture is solved.