The scheme-theoretic linear span of the tangent cone

Choose an affine open $U=\mathrm{Spec}R$ containing $x$, available by [F3], and write $x=\mathfrak{p}$. Then $R$ is Noetherian, and $\mathfrak{p}$ is a submodule of the left regular module $R$ by [F4, F7, F8]. By [F5, F6], it has a finite generating list. The stalk and its maximal ideal are ${\mathcal{O}}_{X,x}=R_{\mathfrak{p}}$ and ${\mathfrak{m}}_x=\mathfrak{p}{R}_{\mathfrak{p}}$ by [F9, F10, F11], so the images of that list generate ${\mathfrak{m}}_x$. Hence $C_{x}X={\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ is finite-dimensional over $k$. By [F2], $T_{x}X$ is its dual, and the finite-dimensional evaluation map $C_{x}X\to (T_{x}X{)}^{\vee }$ is an isomorphism. Thus the coordinate algebra of $\mathbb{A}(T_{x}X)$ is $S_k(C_{x}X)$.

By step 1.1, $S_k(C_{x}X)$ is the coordinate algebra of the affine tangent space. The degree-one quotient $C_{x}X={\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ maps to the degree-one part of ${\mathrm{gr}}_{{\mathfrak{m}}_x}{\mathcal{O}}_{X,x}$ by the identity. Its multiplication extends to a graded algebra map $\varphi :S_k(C_{x}X)\to {\mathrm{gr}}_{{\mathfrak{m}}_x}{\mathcal{O}}_{X,x}$ by [F13, F14]. In degree $d$, every element of ${\mathfrak{m}}_x^d$ is a finite sum of products of $d$ elements of ${\mathfrak{m}}_x$; replacing each factor by its class modulo ${\mathfrak{m}}_x^2$ changes each product only by an element of ${\mathfrak{m}}_x^{d+1}$. Thus $\varphi$ is surjective in every degree. In degree one it is the identity, so if $J=\ker \varphi$, then $J_1=0$.

By [F1], the spectrum of the target of $\varphi$ is ${\mathrm{Cone}}_x(X)$. The surjection $\varphi$ gives a closed immersion ${\mathrm{Cone}}_x(X)\hookrightarrow \mathrm{Spec}{S}_k(C_{x}X)=\mathbb{A}(T_{x}X)$, with scheme ideal $J$. If a linear closed subscheme $L$ contains the cone scheme-theoretically, its ideal is generated by a subspace $W\subseteq C_{x}X$ of degree-one forms and must satisfy $W\subseteq J$. Taking degree-one parts gives $W\subseteq J_1=0$, hence $W=0$ and $L=\mathbb{A}(T_{x}X)$. This proves both the embedding and the claimed scheme-theoretic linear span without replacing the cone by its reduction.

If $C_{x}X=0$, then the target affine space is $\mathrm{Spec}k$ and the surjection in step 2.1 forces every positive graded piece of the local associated graded ring to vanish; the cone is the whole point. For the one-dimensional nonreduced example from [F12] at $x=(ϵ)$, the local ring is $k[ϵ]\mathrm{/}({ϵ}^2)$ and its maximal ideal is $(ϵ)$, so its associated graded ring is $k[ϵ]\mathrm{/}({ϵ}^2)$. The cone is a doubled origin in ${\mathbb{A}}^1$, while its reduction is only the origin; no nonzero linear equation vanishes on the scheme-theoretic cone. The point hypothesis rules out an empty $X$ at the point under discussion. The argument uses one affine neighborhood and a finite generating list for its one prime ideal, and makes no simultaneous choices, basis selection, or Axiom of Choice. The statement is not an iff.