Comparison of two desingularization of the Kontsevich's moduli space of elliptic stable maps

TL;DR

This paper compares two desingularization methods for the main component of Kontsevich's moduli space of elliptic stable maps, showing how the logarithmic stable maps space relates to Vakil-Zinger's desingularization for low degrees.

Contribution

It demonstrates that for degrees up to 3, the moduli space of logarithmic stable maps can be derived from Vakil-Zinger's desingularization via blow-ups.

Findings

Logarithmic stable maps space is obtained by blow-ups of Vakil-Zinger's desingularization for degree ≤ 3.

The main component of the moduli space is singular and requires desingularization.

Comparison clarifies the relationship between two desingularization approaches.

Abstract

It is known that the main component of the Kontsevich's moduli space of elliptic stable maps is singular. There are two different desingularizations. One is Vakil-Zinger's desingularization and the other is the moduli space of logarithmic stable maps. When the degree is less then or equal to 3 and the target is $\mathbb{P}^{n}$, we show that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger's desingularization.