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Daebeom Choi

Publications and source records attributed to Daebeom Choi.

6 recordsLinked to original sources
arXiv2026

Extremal Effective Cycles and Nef Line Bundles on \(\overline{\rm{M}}_{g,n}\)

There has been a growing body of work devoted to the study of effective cones of codimension-\(k\) cycles \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) on \(\overline{\rm{M}}_{g,n}\), the moduli space of \(n\) pointed stable curves of genus \(g\). In this paper, we remove the genus-dependence present in previous bounds on the number of marked points, and prove the following results: (1) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) has infinitely many extremal rays for \(k\ge 2\), \(g\ge 3\) and \(n\ge 2k-2\), and (2) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) is non-polyhedral for \(k\ge 2\), \(g\ge 1\) and \(n\ge k+5\). Moreover, we show that (3) every rational tails boundary stratum spans an extremal ray. Our method refines that of Chen and Coskun by extending arguments based on morphisms, or equivalently semiample divisors, to a setting that also allows for the use of nef divisors. Certain non-semiample nef divisors on \(\overline{\rm{M}}_{g,n}\), namely so-called semigroup kappa divisors of a particular kind, play a crucial role.

math.AG↗
arXiv2025

Extremal effective curves and non-semiample line bundles on $\overline{\rm{M}}_{g,n}$

We develop a new method for establishing the extremality in the closed cone of effective curves on the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by using a general criterion for non-semiampleness which extends Keel's argument, we demonstrate that a substantial portion of the cone of nef divisors of $\overline{\mathrm{M}}_{g,n}$ is not semiample. As an application, we construct the first explicit example of a non-contractible extremal ray of the closed cone of effective curves on $\overline{\mathrm{M}}_{3,n}$. Our method relies on two main ingredients: (1) the construction of a new collection of nef divisors on $\overline{\mathrm{M}}_{g,n}$, and (2) the identification of a tractable inductive structure on the Picard group, arising from Knudsen's construction of $\overline{\mathrm{M}}_{g,n}$.

math.AG↗
arXiv2025

Conformal Block Divisors for Discrete Series Virasoro VOA $\text{Vir}_{2k+1,2}$

In this work, we study a family of vector bundles on the moduli space of curves constructed from representations of $\text{Vir}_{2k+1,2}$, a family of vertex operator algebras derived from the Virasoro Lie algebra. Using the relationship between rank and degree, we characterize their asymptotic behavior, demonstrating that their first Chern classes are nef on $\overline{\rm{M}}_{g,n}$ in many cases. This is the first nontrivial example of divisors arising from vertex operator algebras that are uniformly positive for all genera. Furthermore, for $g = 1$, these divisors form a $\mathbb{Q}$-basis of the Picard group of $\overline{\rm{M}}_{1,n}$, with several desirable functorial properties. Using this basis, we characterize line bundles on certain contractions of $\overline{\rm{M}}_{1,n}$. We also propose conjectures regarding the conformal blocks of Virasoro VOAs and potential generalizations. In particular, by introducing a generalization of conformal block divisors, we provide a nonlinear nef interpolation between affine and Virasoro conformal block divisors.

math.AG↗
arXiv2024

Line bundles on Contractions of $\overline{\rm{M}}_{0,n}$ via Coinvariant Divisors

Using representations of vertex operator algebras, we describe the line bundles on a wide range of contractions of $\overline{\rm{M}}_{0,n}$, the moduli space of stable $n$-pointed rational curves, by proving a stronger version of the contraction theorem for these morphisms. These include the celebrated constructions of Kapranov, Keel, and Knudsen. Our main result suggests that while many so-called F-curves are not $K_X$-negative, they exhibit behavior similar to $K_X$-negative curves. This reveals for instance, a distinguished property of Knudsen's construction $f_{\text{Knu}}:\overline{\rm{M}}_{0,n}\to \overline{\rm{M}}_{0,n-1}\times_{\overline{\rm{M}}_{0,n-2}}\overline{\rm{M}}_{0,n-1}$, allowing for the classification of all possible factorizations of $f_{\text{Knu}}$, as well as further applications, and generalizations.

math.AG↗
arXiv2023

Complete Subvarieties of $\rm{M}_{g,n}$ and a Lifting Problem

Finding the maximal dimension of complete subvarieties of the moduli space of smooth $n$-pointed curves of genus $g$ is a long-standing open problem. Here we show that for $g\ge 3\cdot 2^{d-1}$, if the characteristic of the base field is greater than $2$, then $\rm{M}_g$ contains a complete subvariety of dimension $d$. Furthermore, in positive characteristic, we construct a complete surface in $\rm{M}_{g,n}$ for $g\ge 3$ and $n\ge 1$, which contain a general point. These results follow from the proofs of the lifting conjectures, introduced here. In particular, we translate the existence of complete subvarieties to properties of line bundles on $\rm{M}_{g,n}$. Our method reframes Zaal's approach, with increased efficiency via Keel's results on semi-ample line bundles in positive characteristic. This method demonstrates the difference in the geometry of moduli spaces between characteristic $0$ and characteristic $p$.

math.AG↗
arXiv2021

Non-archimedean Sendov's Conjecture

We prove non-archimedean analogue of Sendov's conjecure. We also provide complete list of polynomials over an algebraically closed non-archimedean field $K$ that satisfy the optimal bound in the Sendov's conjecture.

math.NT↗