Suppose $d>0$, $k$ is a field and $L_d$ is the space of projective plane curves over $k$ of degree $d$, namely $L_d=\mathbb P(k[X,Y,Z]_d)$, where $k[X,Y,Z]_d$ is the vector space of homogeneous polynomials of degree $d$. Let $E\subseteq L_d$ be the collection of curves with at least two $k$-singular points and $F$ be the collection of curves with at least a degenerate $k$-singular point.
- When $k=\mathbb R$, is it true that $E\cup F$ is closed in $L_d$ endowed with classical topology?
- In general, is it true that $E\cup F$ is closed in $L_d$ endowed with Zariski topology?
- What's $\overline E$, the Zariski closure of $E$ in $L_d$?
I can show that $F$ is closed in $L_d$ endowed with classical topology when $k=\mathbb R$ as follows:
Consider the map $\phi\colon S^2\times L_d\to\mathbb R,(x,P)\mapsto\lvert P(x)\rvert+\sum_{j=1}^3\lvert\partial_jP(x)\rvert+\lvert\det\operatorname{Hess}P(x)\rvert$ ($\operatorname{Hess}P$ is the Hessian matrix of $P$), which induces a continuous map $\psi\colon\mathbb{RP}^2\times L_d\to\mathbb R$. Note that $\psi^{-1}(0,\infty)$ is open and if the image of $P$ in $L_d$ lies in $L_d\setminus F$, then $\psi^{-1}(0,\infty)$ contains $\mathbb{RP}^2\times\{[P]\}$, hence by compactness of $\mathbb{RP}^2$, we can conclude that $F$ is closed.
My general framework to attack 1,2,3 is the following idea: let $G=\mathbb{P}^2\times\mathbb{P}^2\setminus\Delta$ where $\Delta$ is the diagonal, we restrict the projection $\pi_2\colon G\times L_d\to L_d$ to a subvariety $S=\{\,(p,q,P)\in G\times L_d\,\vert\,P\textrm{ is singular at }p, q\,\}$. Apparently $E=\pi_2(S)$ but we want to embed $G$ into an appropriate projective variety such that the closure of $S$ in that variety projects to the closure of $E$.
Any ideas? Thanks!