Bound for the number of one-dimensional fibers of a projective morphism

TL;DR

This paper establishes an upper bound on the number of one-dimensional fibers in the canonical projection of the graph of a birational parameterization of an algebraic surface in projective space.

Contribution

It provides a new bound for the number of 1-dimensional fibers in the projection of the graph of a birational surface parameterization.

Findings

Bound on the number of 1-dimensional fibers established

Applicable to birational parameterizations of algebraic surfaces

Enhances understanding of fiber structure in projective morphisms

Abstract

Given a birational parameterization $\phi: \mathbb{P}_k^2 - rightarrow \mathbb{P}_k^3$ of an algebraic surface $\mathscr S\subset \mathbb{P}_k^3$, we bound the number of 1-dimensional fibers of the canonical projection of the graph of $\phi$ onto its image.