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Consider a family of smooth plane conics $f_\lambda(x,y,z)=0$ as a family $T_\lambda = (C,L,v_1,v_2,v_3)_\lambda$ of genus zero curves with a degree 2 line bundle $L$ and an ordered basis $v_i$ for the space of global sections of $L$. Suppose that $f_\lambda$ degenerates to the double line $x^2=0$ when $\lambda=0.$

What are various geometric objects which could serve as a limit of $T_\lambda$ as $\lambda$ goes to zero, and what are some examples of situations where each limit would be (in)appropriate?

For example, the equation $x^2=0$ would be a possible limit that seems appropriate if we're thinking about the $T_\lambda$ as polynomial equations, but less appropriate if we're thinking about the $T_\lambda$ as $5$-ples $(C,L,v_1,v_2,v_3)_\lambda.$

I would be particularly interested in an answer that uses the ribbon structure on $x^2=0.$

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    $\begingroup$ It sounds like you're asking for compactifications of the space of conics. One compactification that is often useful is the space of complete conics. This can be described in various other ways (e.g., as the blowup of $\mathbb{P}^5$ along the Veronese, as a Kontsevich space, etc...) $\endgroup$
    – dhy
    Commented Sep 5, 2021 at 3:24
  • $\begingroup$ Thank you @dhy. I've found some material on this that looks interesting. $\endgroup$
    – Caliper
    Commented Sep 5, 2021 at 18:20
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    $\begingroup$ To add a little bit to @dhy's comment, the space of complete conics is a classical compactification that can be explicitly described as the blowup of $\mathbb P^5$ along the Veronese surface. For plane conics this happens to coincide with a number of other parameter spaces such as the Hilbert scheme and the Kontsevich space of stable maps. Here we can also identify the blow down to $\mathbb P^5$ with the Hilbert-Chow morphism that exists in general between the Hilbert scheme and Chow variety.... $\endgroup$
    – Tabes Bridges
    Commented Sep 6, 2021 at 2:33
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    $\begingroup$ ...However in general the Kontsevich space is distinct from the Hilbert Scheme, so conics are an unusually nice situation. $\endgroup$
    – Tabes Bridges
    Commented Sep 6, 2021 at 2:33

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I am not sure I understand what you are asking, but one possible answer is the following.

The line bundle $L$ on smooth conics is nothing but the restriction of the line bundle $\mathcal{O}(1)$ from $\mathbb{P}^2$. In the same way you can restrict it to the double line, and then it gives you the second part of the limit point data.

Similarly, the space of sections of the restricted line bundle is isomorphic (via the restriction morphism) to the space of sections of $\mathcal{O}(1)$ on $\mathbb{P}^2$, a fixed vector space. The limit of the family of bases thus may or may not exist (because the variety of bases in a given vector space is not proper). But if it does exist, it gives you the last part of the limit point data.

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  • $\begingroup$ Thank you @Sasha. The thought process behind my question is something like this. Suppose we have a family of stable, smooth genus 0 curves $(C, p, q, r_\lambda)$ with $r_\lambda \to q$ as $\lambda \to 0.$ Then we have a process for replacing the unstable fiber above 0 with a stable curve, and we have some good idea for why and what situations we do this in. So there may be circumstances/reasons why we don't use the "obvious" thing as the limit, and I am interested in peoples' thoughts on this. $\endgroup$
    – Caliper
    Commented Sep 5, 2021 at 18:23

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