Skein and cluster algebras of punctured surfaces

We prove the full Fock--Goncharov conjecture for $\mathcal{A}_{SL_2,\Sigma_{g,p}}$, the $\mathcal{A}$-cluster variety of the moduli of decorated twisted $SL_2$-local systems on triangulable surfaces $\Sigma_{g,p}$ with at least 2 punctures. Equivalently, we show that the tagged skein algebra $Sk^{ta}(\Sigma)$, or the middle cluster algebra $\mathrm{mid}(\mathcal{A})$, coincides with the upper cluster algebra $U(\Sigma)$. Inspired by the work of Shen--Sun--Weng, we introduce the localized cluster variety $\mathring{\mathcal{A}}$ as the algebraic version of the decorated Teichm\"uller space $\mathcal{T}^d(\Sigma)$. We show its global section $\Gamma(\mathring{\mathcal{A}},\mathcal{O}_{\mathring{\mathcal{A}}})$ equals the classical Roger--Yang skein algebra $Sk^{RY}_{q\to1}(\Sigma)$, thereby providing a quantization of $\mathcal{T}^d(\Sigma)$ in terms of the Roger--Yang skein algebra $Sk^{RY}_q(\Sigma)$. As a consequence of our geometric characterizations, we deduce normality and the Gorenstein property of the tagged skein algebra $Sk^{ta}(\Sigma)$ and the classical Roger--Yang skein algebra $Sk^{RY}_{q\to1}(\Sigma)$, as well as finite generation of upper cluster algebra $U(\Sigma)$.