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Let us consider the family of Fermat conics in $(\mathbb{C}^*)^2\subset\mathbb{C}^2$ given by $$\pi\colon V(ax^2+by^2-1)\subset(\mathbb{C}^*)^2_{a,b}\times(\mathbb{C}^*)^2_{x,y}\to(\mathbb{C}^*)^2_{a,b}$$ We know that $\pi$ does not admit rational sections: the generic conic has index 2.

Taking tropicalization functor, we get the morphism $$\mathrm{Trop}(\pi)\colon \mathrm{Trop}(ax^2+by^2-1)\subset\mathbb{R}^4\to\mathbb{R}^2$$

Question. Does $\mathrm{Trop}(\pi)$ admit tropical rational sections (namely, sections over $\mathbb{R^2}\backslash W$ for some proper tropical subvariety $W\subset\mathbb{R}^2$)?

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  • $\begingroup$ Do you mean to replace $\mathbb{C}$ by some valued field? Otherwise all the fibres will look like a tropical line with multiplicity 2 on every edge. $\endgroup$
    – Yoav Len
    Commented May 26, 2021 at 12:50

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