Positivity of Intersection Multiplicity Over a Two-Dimensional Base

TL;DR

This paper proves Serre's Intersection Multiplicity Conjecture for certain two-dimensional local rings and extends the result to schemes over such bases, linking intersection multiplicity with transversality.

Contribution

It establishes the conjecture for formal power series rings over two-dimensional regular local rings and relates intersection multiplicity to transversality in unramified cases.

Findings

Proves Serre's conjecture in the two-dimensional setting.

Extends results to schemes over smooth two-dimensional bases.

Analyzes the connection between intersection multiplicity and transversality.

Abstract

We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which is a smooth extension of a regular, two-dimensional base Y. We also investigate the connection between intersection multiplicity and transversality in the unramified setting via a local analysis on the blowup.