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Munchlax
Munchlax
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A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a birationalrational map

$H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4\dashrightarrow {\rm Hilb}^6\mathbb{P}^2$${\rm Hilb}^6\mathbb{P}^2\dashrightarrow H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4$

Given a 1-parameter family $X_t$$Z_t$ of smooth cubic surfaceslength six subschemes of $\mathbb{P}^2$, where $Z_t$ is reduced for $t\neq 0$, specializing to $X_0$$Z_0$, how dowhat happens to the six points degeneratesingularities of the corresponding 1-parameter family $X_t$ of cubic surfaces? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.

A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a birational map

$H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4\dashrightarrow {\rm Hilb}^6\mathbb{P}^2$

Given a 1-parameter family $X_t$ of smooth cubic surfaces specializing to $X_0$, how do the six points degenerate? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.

A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a rational map

${\rm Hilb}^6\mathbb{P}^2\dashrightarrow H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4$

Given a 1-parameter family $Z_t$ of length six subschemes of $\mathbb{P}^2$, where $Z_t$ is reduced for $t\neq 0$, specializing to $Z_0$, what happens to the singularities of the corresponding 1-parameter family $X_t$ of cubic surfaces? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.

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Munchlax
Munchlax
  • 323
  • 1
  • 6

Cubic surfaces and configurations of 6 points

A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a birational map

$H^0(\mathbb{P}^3,\mathscr{O}_{\mathbb{P}^3}(3))//PGL_4\dashrightarrow {\rm Hilb}^6\mathbb{P}^2$

Given a 1-parameter family $X_t$ of smooth cubic surfaces specializing to $X_0$, how do the six points degenerate? For example, I know that $X_0$ having an $A_1$ singularity could be from 3 points becoming collinear or 6 points lying on a conic (contributing to a (-2) curve getting collapsed under the canonical map), but I don't know of references for other singularities.