# Zariski closure and complex curve selection

Let $$X\subset \mathbb{C}^n$$ be a quasi affine variety, and let $$x\in \overline{X}\setminus X$$. By the real analytic curve selection lemma, there exists a real analytic curve $$\gamma:[0,\epsilon)\to \bar{X}$$ with $$\gamma(0)=x$$ and $$\gamma((0,\epsilon))\subset X$$.

My question is if is also true that there exists a map $$Spec(\mathbb{C}[[t]])\to \overline{X}$$ satisfying the analogous properties.

• If I've got my terminology right, a very, very special case of this is Lemma 3.1 and Theorem 3.4 of Kempf - Instability in invariant theory. Sep 8 '20 at 12:50
• What am I missing? I would think that there has to be a complex algebraic curve contained in $\bar X$, not contained in $X$, containing the point $x$; and that normalizing this would provide not just a formal curve but a complex analytic curve with the analogous property.. Nov 1 '20 at 1:40
• @TomGoodwillie You are not missing anything, this is I think the most concise argument to show the claim. Nov 1 '20 at 12:32

Depending possibly on exactly how you define everything I think the answer is yes in a more general situation:

$$\textbf{Proposition:}$$ Let $$\overline{X}$$ be a finite type scheme over $$\mathbb{C}$$, and $$X \subset \overline{X}$$ a dense open, and $$x \in \overline{X} \setminus X$$ a closed point. Then there exists a map $$\mathrm{Spec}(\mathbb{C}[[t]]) \to \overline{X}$$ such that $$(t) \mapsto x$$ and $$(0) \mapsto y$$ for some $$y \in X$$ which will be the generic point of a curve passing through $$x$$.

Notice that any map $$\mathrm{Spec}(\mathbb{C}[[t]]) \to \overline{X}$$ factors through $$\mathrm{Spec}(\mathbb{C}[[t]]) \to \mathrm{Spec}(\mathcal{O}_{X,x}) \to \overline{X}$$ where $$\mathcal{O}_{X,x} \to \mathbb{C}[[t]]$$ is a local map (such that $$(t) \mapsto x$$). Let $$A = \mathcal{O}_{X,x}$$.

There is a nonempty principal open $$D(f) \subset X \cap \mathrm{Spec}(\mathcal{O}_{X,x})$$ for some $$f \in A$$ with $$f \notin \mathrm{nilrad}(A)$$ but $$x \notin D(f)$$ since $$x \in \overline{X} \setminus X$$. Now we apply the following lemma:

$$\textbf{Lemma:}$$ Let $$(A, \mathfrak{m})$$ be a Noetherian local ring with finite dimension $$\mathrm{dim}(A) \ge 1$$ and let $$f \in \mathfrak{m}$$ with $$f \notin \mathrm{nilrad}(A)$$. Then there exists a prime ideal $$\mathfrak{p} \subset A$$ such that $$f \notin \mathfrak{p}$$ and $$\mathrm{dim}(A / \mathfrak{p}) = 1$$.

Proof: there is a prime $$\mathfrak{p}_0 \subset A$$ with $$f \notin \mathfrak{p}_0$$ because $$f \notin \mathrm{nilrad}(A)$$. Therefore replacing $$A$$ by $$A / \mathfrak{p}_0$$ we can assume $$A$$ is a domain. Now we proceed by induction on $$\mathrm{dim}(A) = n$$. If $$\mathrm{dim}(A) = 1$$ then we have prime ideals $$(0) \subsetneq \mathfrak{m}$$ so taking $$\mathfrak{p} = (0)$$ the result follows. Otherwise, choose a maximal length chain of primes, $$(0) \subsetneq \mathfrak{p}_1 \subsetneq \mathfrak{p}_2 \subsetneq \cdots \subsetneq \mathfrak{p}_n = \mathfrak{m}$$ Then there are infinitely many prime ideals $$\mathfrak{p}_1'$$ strictly between $$(0)$$ and $$\mathfrak{p}_2$$ which all must have height one since this chain is maximal length (Kaplansky, Commutative Rings, Thm. 144). And thus some $$\mathfrak{p}_1'$$ does not contain $$f$$ (Kaplansky, Commutative Rings). Therefore, $$A' = A / \mathfrak{p}_1'$$ satisfies the hypotheses and $$\mathrm{dim}(A') = n - 1$$ so we conclude.

Returning to the propositon, we get a prime $$\mathfrak{p} \in D(f) \subset X$$ with $$B = A / \mathfrak{p}$$ a Noetherian local domain of dimension one. Let $$C$$ be the integral closure of $$B$$ in $$\mathrm{Frac}(B)$$ so $$B \subset C$$ is an integral extension of Noetherian domains and thus $$\mathrm{dim}(C) = 1$$ and $$\mathrm{Spec}(C) \to \mathrm{Spec}(B)$$ is surjective so there is a maximal ideal $$\mathfrak{m}' \subset C$$ above $$\mathfrak{m}$$ and thus $$B \to C_{\mathfrak{m}'}$$ is a local inclusion of Noertherian domains where $$C_{\mathfrak{m}'}$$ is a DVR since it is integrally closed. Then $$\widehat{C_{\mathfrak{m}'}} \cong \mathbb{C}[[t]]$$ by the Cohen structure theorem. So we get a local injection $$B \to C \to C_{\mathfrak{m}'} \to \mathbb{C}[[t]]$$. Thus the map, $$\mathrm{Spec}(\mathbb{C}[[t]]) \to \mathrm{Spec}(\mathcal{O}_{X,x}/\mathfrak{p}) \to \mathrm{Spec}(\mathcal{O}_{X,x}) \to \overline{X}$$ sends $$(t) \mapsto x$$ and $$(0) \mapsto \mathfrak{p} \in D(f)$$.