On the self-intersection number of the nonsingular models of rational cuspidal plane curves

TL;DR

This paper establishes an upper bound for the self-intersection number of proper transforms of rational cuspidal plane curves with at least three cusps, characterizing the quartic curve with three cusps as the unique case attaining this bound.

Contribution

It provides a new upper bound for the self-intersection number of these curves and characterizes the quartic with three cusps as the unique maximizer.

Findings

The self-intersection number is bounded above by a specific value.

The quartic curve with three cusps uniquely attains this bound.

The bound is achieved if and only if the curve is the quartic with three cusps.

Abstract

In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a curve having exactly three cusps, we show that the self-intersection number is equal to the bound if and only if the curve coincides with the quartic curve having three cusps.