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Consider a cubic surface cut out by equations $x^2y - z^2w$ inside $\mathbb{P}^3$. This gives a cubic surface with a line of nodes, it is toric and has normalisation $\mathbb{F}_1$, a Hirzebruch surface.

My confusion is two fold, I am having difficulty spotting which two torus invariant lines of $\mathbb{F}_1$ are being glued together.

In addition I would like to understand how this degeneration works with respect to the 27 lines on a cubic surface and see where degenereate to.

** Edit:** Sasha's answer has made me delete some of my post as it has clarified why it is wrong, and now it is unhelpful and long.

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The cubic scroll in $\mathbb{P}^4$ is isomorphic to $\mathbb{F}_1$ and its torus-invariant divisor has three line components and one conic component. The linear projection $\mathbb{P}^4 \dashrightarrow \mathbb{P}^3$ from a point lying in the linear span of the conic, but not on the conic itself, gives the normalization map $$ \mathbb{F}_1 \to X = \{x^2y - z^2w = 0\} \subset \mathbb{P}^3. $$ Thus, instead of gluing two lines, it collapses the conic onto a line via a 2:1 map.

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  • $\begingroup$ Ok it seems like this will answer my question. Is there any way of seeing this from the toric geometry? $\endgroup$ – UserUser Nov 18 '19 at 9:00
  • $\begingroup$ I guess with embedding $a^2b - c^2d$ and $bd=e^2$ there should be a way to see this map. $\endgroup$ – UserUser Nov 18 '19 at 9:15

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