arXiv subjects

Keli Santos-Parker

Publications and source records attributed to Keli Santos-Parker.

2 recordsLinked to original sources

arXiv2019

Moduli of stable maps in genus one and logarithmic geometry II

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus $1$.

math.AG↗

arXiv2019

Moduli of stable maps in genus one and logarithmic geometry I

This is the first in a pair of papers developing a framework for the application of logarithmic structures in the study of singular curves of genus $1$. We construct a smooth and proper moduli space dominating the main component of Kontsevich's space of stable genus $1$ maps to projective space. A variation on this theme furnishes a modular interpretation for Vakil and Zinger's famous desingularization of the Kontsevich space of maps in genus $1$. Our methods also lead to smooth and proper moduli spaces of pointed genus $1$ quasimaps to projective space. Finally, we present an application to the log minimal model program for $\mathcal{M}_{1,n}$. We construct explicit factorizations of the rational maps among Smyth's modular compactifications of pointed elliptic curves.

math.AG↗