The Stack of Rational Nodal Curves

TL;DR

This paper constructs and stratifies the stack of rational nodal curves of genus 0, revealing its structure and properties, including non-representability of the universal curve map, as part of a series studying its Chow ring.

Contribution

It introduces the stack of rational nodal curves, details its stratification, and demonstrates the non-representability of the universal curve map, advancing understanding of its geometric structure.

Findings

Constructed the stack of rational nodal curves.

Established stratification by number of nodes.

Proved the universal curve map is not representable.

Abstract

In this series of three papers we start to investigate the rational Chow ring of the stack consisting of nodal curves of genus 0, in particular we determine completely the rational Chow ring of the substack consisting of curves with at most 3 nodes. In this first paper we construct the stack of rational nodal curves and its stratification by nodes and show that the map from the universal curve to the stack is not representable in the category of schemes.