Research

  1. Line bundles on fantastacks and their cohomology. (2026)

    Abstract

    Let $\mathcal{F}_{\Sigma,\beta}$ be a fantastack determined by $\beta \colon \mathbb{Z}^n \to N$ and a fan $\Sigma \subseteq N_{\mathbb R}$, and $\mathscr{L}$ be a line bundle. We construct a fan $\Sigma'$ and a torus-invariant Cartier divisor $D'$ such that for all $i \geq 0$, we have

    \[ H^i(\mathcal{F}_{\Sigma,\beta},\mathscr{L}) \cong H^i(X_{\Sigma'}, \mathscr O(D')). \]

Writings

  2. A survey of toric stacks. (2026)

    Abstract

    In this report, we study toric stacks, a rich class of quotient stacks governed by combinatorics. We pursue a self-contained approach, developing the theory through examples. Our main result is the following theorem.

    Theorem. For any fantastack $\mathcal{F}_{\Sigma,\beta}$, we have that

    \[ \operatorname{Pic}(\mathcal{F}_{\Sigma,\beta}) \cong \operatorname{coker}(\beta^\vee). \]

    Moreover, for any fan $\Sigma$, we have

    \[ \operatorname{Pic}(\mathcal{F}_{\Sigma,\beta_{\Sigma}}) \cong \operatorname{Cl}(X_\Sigma). \]
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