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This is a Quantitative Study study.

On compactifications of the 𝑆𝐿(2,ℂ) character varieties of punctured surfaces

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Boundary divisors in the character varieties are confirmed to be toric varieties, advancing current understanding.
Explicit results were obtained regarding projective compactifications based on ideal triangulations of punctured surfaces.
The method improves upon Komyo's previous approach to compactifying relative character varieties.
The construction of a uniform relative compactification over C^n contributes to ongoing questions in the field.

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This paper investigates boundary divisors and triangulations in character varieties, indicating new compactification methods.

Tehrani et al. (2025) studied this question.

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