An irreducible curve can have arbitrarily large tangent dimension

The image $R=\psi (k[x_n,\dots ,x_{2n-1}])$ of the unique $k$-algebra map $\psi$ with $x_i\mapsto t^i$ given by [F2] is a subring of $k[t]$ and a finite-type $k$-algebra, and [F3] identifies $R$ with $k[x_n,\dots ,x_{2n-1}]\mathrm{/}\ker \psi$; by [F5, F6] the ring $k[t]$ is a domain, hence so is its subring $R$, and $C=\mathrm{Spec}R$ is a $k$-scheme by [F18].

The subring description: every element of $R$ is a $k$-linear combination of products of the generators $t^n,\dots ,t^{2n-1}$ by [F2, F3], and every nonempty such product is $t^{i_1+\cdots +i_r}$ with $i_1+\cdots +i_r\ge n$ by [F1], so $R\subseteq k\cdot 1+{\mathrm{span}}_k\{t^j:j\ge n\}$; conversely $1\in R$ and $t^j\in R$ for every $j\ge n$, by strong induction on $j$ ($t^j$ is a generator for $n\le j\le 2n-1$, and $t^j=t^{j-n}{t}^n$ with $j-n\ge n$ for $j\ge 2n$); hence $R=k\cdot 1\oplus {\mathrm{span}}_k\{t^j:j\ge n\}$ with the sum direct and coefficients unique by [F4].

Suppose for contradiction that $\iota :C\to {\mathbb{A}}_k^{n-1}$ is a closed immersion of $k$-schemes, where ${\mathbb{A}}_k^{n-1}=\mathrm{Spec}k[x_1,\dots ,x_{n-1}]$ by [F1, F18]. By [F22] and [F23], the induced map on local rings is surjective at every source point.

The origin and irreducibility: the constant-coefficient map $\epsilon :R\to k$ is a surjective $k$-algebra homomorphism whose kernel is $\mathfrak{m}={\mathrm{span}}_k\{t^j:j\ge n\}=(t^n,\dots ,t^{2n-1})R$ by step 2.1, so [F3] gives $R\mathrm{/}\mathfrak{m}\cong k$, making $\mathfrak{m}$ maximal with residue field $k$ and exhibiting $0:=\mathfrak{m}$ as the $k$-rational point $\epsilon$ of $C$ under [F20]; since $R$ is a domain by step 1.1, the zero ideal is prime and [F11] applied to the nonempty closed subset $\mathrm{Spec}R=V(0)$ shows that $C$ is irreducible.

The dimension is one: $t$ is integral over $R$ because it is a root of the monic polynomial $X^n-t^n\in R[X]$ by [F7], and the integral elements of $k[t]$ over $R$ form a subring by [F8] containing $R$ and $t$, hence containing $R[t]=k[t]$; thus $R\subseteq k[t]$ is an injective integral extension of nonzero commutative rings and $\dim R=\dim k[t]=1$ by [F9] and [F10], with the Axiom of Choice entering exactly through [F9].

The tangent directions: ${\mathfrak{m}}^2={\mathrm{span}}_k\{t^s:s\ge 2n\}$, because a product of two elements of $\mathfrak{m}$ is a $k$-linear combination of monomials $t^{j+l}$ with $j,l\ge n$ and hence lies in that span, while conversely $t^s=t^{s-n}\cdot t^n\in {\mathfrak{m}}^2$ for every $s\ge 2n$ by step 2.1; therefore $\mathfrak{m}={\mathrm{span}}_k\{t^n,\dots ,t^{2n-1}\}\oplus {\mathfrak{m}}^2$, and the classes $b_i:=t^i+{\mathfrak{m}}^2$ for $n\le i\le 2n-1$ form a $k$-basis of the cotangent space $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ by [F12, F16, F4].

The points: $0\in C(k)$ by step 3.1 composes with $\iota$ to a $k$-rational point $p=\iota \circ 0$ of ${\mathbb{A}}_k^{n-1}$ by [F18, F19]. Under the anti-equivalence [F20] the point $p$ corresponds to a $k$-algebra map $k[x_1,\dots ,x_{n-1}]\to k$, that is, to a tuple $a\in k^{n-1}$ with maximal ideal $(x_1-a_1,\dots ,x_{n-1}-a_{n-1})$ by [F2, F21].

The tangent dimension: $T_{0}C={\mathrm{Hom}}_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2,k)$ is a $k$-vector space by [F13, F14, F12], and for each $i$ the coordinate functional $b_i^{\ast }$ of [F15] is an element of it; every $\lambda \in T_{0}C$ satisfies $\lambda ={\sum }_i\lambda (b_i)b_i^{\ast }$ because both sides are $k$-linear and agree on the basis $(b_i)$ of [F16], and ${\sum }_i{c}_i{b}_i^{\ast }=0$ forces $c_j=0$ by evaluation at $b_j$; hence $(b_n^{\ast },\dots ,b_{2n-1}^{\ast })$ is a $k$-basis of $T_{0}C$ with $n$ elements and ${\dim }_k{T}_{0}C=n$ by [F17].

The contradiction: by step 2.2 and [F23], the stalk map ${\mathcal{O}}_{{\mathbb{A}}^{n-1},p}\twoheadrightarrow {\mathcal{O}}_{C,0}$ is a surjective local $k$-homomorphism. It maps the maximal ideal ${\mathfrak{m}}_p$ onto ${\mathfrak{m}}_0$, hence maps ${\mathfrak{m}}_p^2$ onto ${\mathfrak{m}}_0^2$ and induces a surjection ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2\twoheadrightarrow {\mathfrak{m}}_0\mathrm{/}{\mathfrak{m}}_0^2$. Dualizing over the common residue field $k$ yields an injection $T_{0}C\hookrightarrow T_p{\mathbb{A}}_k^{n-1}$ by [F12, F13]. Applying [F24] to the affine space with actual ideal $(0)$ gives $T_p{\mathbb{A}}_k^{n-1}\cong \ker (0:k^{n-1}\to 0)=k^{n-1}$; thus [F25] and [F26] imply ${\dim }_k{T}_{0}C\le n-1$. But step 5.1 gives ${\dim }_k{T}_{0}C=n$, a contradiction. Therefore no closed immersion $C\to {\mathbb{A}}_k^{n-1}$ exists.

Conclusion: for every $n\ge 1$ the monomial curve $C=\mathrm{Spec}k[t^n,\dots ,t^{2n-1}]$ is irreducible of dimension one (steps 3.1 and 3.2), its tangent space at the $k$-rational origin has dimension exactly $n$ (step 5.1), and it admits no closed immersion into ${\mathbb{A}}_k^{n-1}$ (step 6.1); the tangent dimension of an irreducible curve is therefore unbounded, completing the source's exercise with an irreducible witness.

Boundary and scope dispositions: $C$ is nonempty and irreducible with the $k$-rational point $0$ by step 3.1, and ${\mathbb{A}}_k^0=\mathrm{Spec}k$ is nonempty, so no boundary object is empty; the zero cases are the zero ideal of $k[t]$ used in step 3.1 and the zero vector of each $k$-vector space, and for $n$ as small as possible the ideal $\mathfrak{m}$ is nonzero and principal-generated by $t$; the minimal instance $n=1$ gives $R=k[t]$, $C={\mathbb{A}}_k^1$, $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2=k\cdot t$ of dimension $1$, with the closed-embedding obstruction against ${\mathbb{A}}_k^0$ still delivered by step 6.1; the degenerate case of the construction is this same $n=1$ endpoint, where the subring is all of $k[t]$, the integral extension of step 3.2 is an isomorphism and no higher-degree gap survives, while for $n\ge 2$ the degree gap $[n,2n-1]$ between the least element of $\mathfrak{m}$ and the least element of ${\mathfrak{m}}^2$ is exactly the source of the $n$ independent directions in steps 4.1 and 5.1, its endpoint being $2n-1$; every construction is canonical, the only choice-theoretic inputs being the explicitly declared suppliers [F9, F11] named in steps 3.1 and 3.2, while the tuple $a$ is determined by the rational point $p$, and no other simultaneous or dependent choice occurs; and no biconditional is asserted as a claim, the only equivalences used being the anti-equivalence of [F20], the irreducibility criterion of [F11] applied in one direction and the defining sheaf-surjectivity of [F23] applied in one direction, so the iff axes are vacuous here.