# Log Calabi-Yau surfaces without maximal boundaries

Let $$X$$ be a smooth projective surface over $$\mathbb{C}$$, $$D\subset X$$ is an effective divisor. $$(X,D)$$ is a log Calabi-Yau pair if $$K_X+D$$ is a principal divisor. The complement $$M=X\setminus D$$ is a log Calabi-Yau surface if the curve $$D$$ has at worst nodal singularities. The pair $$(X,D)$$ is said to have maximal boundary if $$D$$ is actually nodal. As a consequence of Bertini's theorem, one can always find a smoothing $$\widetilde{D}$$ of $$D$$ in the same linear system so that $$(X,\widetilde{D})$$ becomes a log Calabi-Yau pair without maximal boundary. However, I'm interested in whether the converse is true, namely whether starting from a log Calabi-Yau pair $$(X,\widetilde{D})$$ such that $$\widetilde{D}$$ is not maximal, one can always find a degeneration $$D$$ of $$\widetilde{D}$$ so that $$(X,D)$$ is a log Calabi-Yau pair with maximal boundary?

• Why you can always find a smoothing? The linear system is not necessarily base point free. Jul 7, 2019 at 2:24

Firstly, I do not think that every maximal boundary has a smoothing. For example, if the linear system $$|-K_X|$$ as fixed part, then very likely it contains no smooth element.
On your question of the converse part, we can not expect to find a maximal boundary. For the easiest example, just consider $$X=E\times \mathbb{P}^1$$, where $$E$$ is an elliptic curve. Then every element in the linear system of $$-K_X$$ Is just two copies of $$E$$, which is smooth. So there is mo nodal boundary.
• ...or an abelian surface (with $D=0$). Maybe the question becomes more interesting if $X$ is rationally connected (or Fano?). Jul 7, 2019 at 10:21