Not every abelian variety over a Dedekind function field extends to an abelian scheme

The curve $E_t$ is an elliptic curve over $K$ by the chord-tangent construction, and $x^3+t$ has no root in $K=k(\text{\hspace{0.17em}⁣}(t)\text{\hspace{0.17em}⁣})$: a root would satisfy $3v(x)=v(t)=1$ for the valuation $v$, impossible for an integer valuation. Hence $E_t[2](K)=\{O\}$ by [F2], a set with one element.

Suppose there were an abelian scheme model $A\to \mathrm{Spec}R$ with generic fibre $E_t$. By [F1] multiplication by $2$ is finite etale of rank $4$ on $A$, so $A[2]$ is finite etale over $R$; properness of $A$ extends each $K$-point of $A[2]$ uniquely to $R$, and finite etale lifting identifies $A[2](R)$ with $A[2](k)$. The special fibre is an elliptic curve over $k$ (characteristic not $2$), so $A[2](k)=\mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2$ has four elements by the multiplication theorem, whence $A[2](K)=E_t[2](K)$ would have four elements, contradicting step 1.1.

This contradiction proves that no abelian scheme model of $E_t$ over $R$ exists: the elliptic curve $E_t$ has bad reduction. The companion example records the explicit ramified extension $k(\text{\hspace{0.17em}⁣}(s)\text{\hspace{0.17em}⁣})$ with $s^6=t$ over which the curve acquires good reduction, so the absence of a model over $R$ is genuinely a failure of descent and not an artifact of the chosen equation.