D0-brane realizations of the resolution of a reduced singular curve

TL;DR

This paper demonstrates that resolutions of reduced singular curves can be realized through D0-brane configurations, linking string theory concepts with algebraic geometry and proving a conjecture for curves.

Contribution

The paper proves a conjecture that any resolution of a reduced singular curve can be embedded into a stack of punctual D0-branes, extending the understanding of D-brane realizations of singularity resolutions.

Findings

Resolution of a reduced singular curve can be obtained via D0-brane configurations.

The rank of D0-branes needed can be arbitrarily large.

The conjecture holds specifically for reduced singular curves.

Abstract

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space $Y$ via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution $Y^{\prime}\rightarrow Y$ of a variety $Y$ over ${\Bbb C}$ can be factored through an embedding of $Y^{\prime}$ into the stack ${\frak M}^{0^{A z^f}_{\;p}}_r (Y)$ of punctual D0-branes of rank $r$ on $Y$ for $r\ge r_0$ in ${\Bbb N}$, where $r_0$ depends on the germ of singularities of $Y$. We prove that this conjecture holds for the resolution $\rho: C^{\prime}\rightarrow C$ of a reduced singular curve $C$ over ${\Bbb C}$. In string-theoretical language, this says that the resolution $C^{\prime}$ of a singular curve $C$ always arises from an appropriate D0-brane aggregation on $C$ and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.