Flash Geometry of Algebraic Curves

TL;DR

This paper develops the theory of flashes for algebraic curves, establishing its birational invariance and applying it to asymptotic degenerations, including a correction to Severi's conjecture proof.

Contribution

It introduces the concept of flashes for algebraic curves and demonstrates their invariance, providing a new foundation for degenerations and correcting a classical conjecture.

Findings

Theory of flashes is birationally invariant

Provides a foundation for asymptotic degenerations

Corrects Severi's original proof of his conjecture

Abstract

In this paper, we develop the theory of flashes of an algebraic curve. We show that the theory is birationally invariant in a sense which we will make more precise below. We also show how the theory provides a foundation for the method of asymptotic degenerations, a particular class of degenerations of plane projective algebraic curves. In particular, we consider the geometrical technique in relation to the Severi problem of degenerating nodal curves to lines in general position, and correct Severi's original proof of his conjecture.