## STNB2022 (35th edition)

### On a special fiber for plane curves.

#### Presenters

Francesc Bars Cortina

#### Abstract

Take $K$ a field with a discrete valuation, and $\mathcal{O}_K$ its ring of integers an fix $f=\sum_{i,j} a_{i,j} xî y^j\in K[x,y]$ a plane curve with $C_0: f=0$ a smooth curve in $\mathbb{G}_{m,K}^2$. By taking the closure of $f=0$ in certain toric scheme one can construct a $\mathcal{O}_K$-model $\mathcal{C}$ for the completion of $C_0/K$ . The core of the talk consists in presenting the fiber of such $\mathcal{O}_K$-model $\mathcal{C}$, and provide some concrete examples.