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Let $C$ be an elliptic curve over $k=\mathbb{C}$ and $\mathcal{L}$ a line bundle of degree $d$. It induces naturally a $\mathbb{A}^1$-fibration $L \to C$ where $L=\underline{\operatorname{Spec}} (\operatorname{Sym}^{\bullet}(\mathcal{L}))$.

We can naturally embed $C$ as zero section $C_0\subset L$ corresonding to canonical surjection $\operatorname{Sym}^{\bullet}(\mathcal{L}) \to \mathcal{O}_C$ sending all positive degree elements of the symmetric algebra to zero.

Q: How to see that the self-intersection number $C_0^2=\text{deg}_L \mathcal{O}(C_0) \vert _{C_0}$ equals $d$? Primary I was interested in 'pure algebraic' proof, but if using analytic methods allow simplifications, I'm eager to see it also, so let me formulate this question more precisely as an "algebraic-vs-analytic-proof comparison" of the above claim.

Source: Example 2.5 in this paper by P. Wagreich on page 425.

# Exit/Remark on objections on compactness issues in the comment: As Damian Rössler noticed it is usual ( ...or at least in "classical" setting) to define intersection numbers in compact/ proper setting, therefore I'm not sure if the "line bundle" $L$ I introduced above as $\underline{\operatorname{Spec}} (\operatorname{Sym}^{\bullet}(\mathcal{L}))$ ( which is obviouly not compact/proper) is that one which Wagreich considered to define the self-intersection number $C_0^2$ or if he tacitly considered instead the "projectivised" version $\mathbb{P}(\mathcal{L} \oplus \mathcal{O}_C)$ instead as $L$...

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    $\begingroup$ The self-intersection is only defined in general if the ambient variety is complete. However $L$ is not complete so you should explain what you want to compute. $\endgroup$
    – Damian Rössler
    Commented Oct 17, 2023 at 15:53
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    $\begingroup$ @DamianRössler It seems pretty clear to me: The intersection is well-defined as a cycle class on the set-theoretic intersection. The set-theoretic intersection is complete so you can take the degree. $\endgroup$
    – Will Sawin
    Commented Oct 17, 2023 at 16:17
  • $\begingroup$ @DamianRössler: Thanks, a good catch, I see to problem, because otherwise the intersection might be not well defined, since some points could "vanish" if we pass the an equivalent divisor. Hmm, the source where I found this example is this paper example 2.5 at page 425. I'm not sure precisely what Wagreich meant there by a line bundle $L \to X$ where he defined this intersection number. Maybe - that's just a guess of mine - he wanted to consider tacitly the "compactified" version $L=\mathbb{P}(\mathcal{L} \oplus \mathcal{O}_X)$ $\endgroup$
    – user267839
    Commented Oct 17, 2023 at 16:19
  • $\begingroup$ of this line bundle... otherwise without compactification as you said one might run into troubles with the well definedness of intersection numbers... $\endgroup$
    – user267839
    Commented Oct 17, 2023 at 16:21
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    $\begingroup$ @DamianRössler No, I am referring to the intersection theory as in Fulton's book, where one does not define the intersection product of two rational equivalence classes of cycles but instead really of two cycles. In this theory the intersection is an equivalence class on the intersection of the supports of the original cycles, which is well-defined since we are intersecting honest cycles. This construction will agree with the intersection number in a compactified setting. $\endgroup$
    – Will Sawin
    Commented Oct 17, 2023 at 18:59

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The result in question is a special case of the self-intersection formula (Fulton, intersection theory, p. 103) which states that the intersection of a smooth subvariety $X$ of a smooth variety $Y$ with itself is the top Chern class of the normal bundle of $X$ in $Y$.

Here the intersection is defined as a certain cycle class in the set-theoretic intersection of the two varieties to be intersected, i.e. in $X$. If $X$ has dimension half the dimension of $Y$, this gives a zero-dimensional class, and if the $X$ is proper, then we can take the degree of this class, which gives the intersection number.

So indeed the intersection number is the degree of the top Chern class. For a curve in a variety, this shows the intersection number is just the degree of the normal bundle.

This is a theorem, not a definition. The definition is on p. 92.

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