Expected number and distribution of critical points of real Lefschetz pencils

TL;DR

This paper develops an asymptotic probabilistic formula to compute the expected number and distribution of critical points of random real Lefschetz pencils on smooth real algebraic varieties, extending classical Riemann-Hurwitz results.

Contribution

It introduces a new asymptotic probabilistic approach to analyze critical points of real Lefschetz pencils, generalizing the Riemann-Hurwitz formula to higher dimensions.

Findings

Derived an asymptotic formula for expected real ramification indices

Computed the expected number of critical points of random real Lefschetz pencils

Analyzed the distribution of critical points on real algebraic varieties

Abstract

We give an asymptotic probabilistic real Riemann-Hurwitz formula computing the expected real ramification index of a random covering over the Riemann sphere. More generally, we study the asymptotic expected number and distribution of critical points of a random real Lefschetz pencil over a smooth real algebraic variety.