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For $X\subseteq {\mathbb{A}}^n$ and $P\in X$, the Zariski tangent space isEquivalently, it is the kernel of the Jacobian of generators of $I(X)$ at $P$. With the tangent-space definition used here,and $P$ is smooth when $\dim T_{X,P}=\dim X$.

Now let $X=Z(f)$. Write $g$ for the square-free product of the distinct irreducible factors of $f$. Hilbert's Nullstellensatz givesThus $T_{X,P}=\ker (d{g}_P)$, whose dimension is at least $n-1$. There is some $P\in X$ with $d{g}_P\ne 0$. Otherwise every partial derivative of $g$ would vanish on $X$, hence belong to $(g)$ by the Nullstellensatz. Its degree is smaller than $\mathrm{deg}g$, so every partial derivative would be zero; in characteristic zero this would make $g$ constant, a contradiction. At such a point the tangent space has dimension $n-1$, and therefore

Next suppose $f$ is irreducible of degree at least two and $P$ is smooth. Translate $P$ to the origin and writeThen $H=T_{X,P}=Z(\ell )$ is a hyperplane. The restriction $f{|}_H$ is nonzero: otherwise $\ell$ would divide $f$, contradicting irreducibility and $\mathrm{deg}f\ge 2$. Since $f(P)=0$, the restriction is nonconstant, so the first part givesFor the natural intersection cut out by $(f,\ell )$, the two differentials at $P$ are $d{f}_P=\ell$ and $d\ell =\ell$. Hence they impose only one independent tangent equation andTherefore $X\cap T_{X,P}$ is singular at $P$.

Finally, represent $\phi :{\mathbb{A}}^2\to {\mathbb{A}}^3$ byIt is not injective exactly when $\mathrm{rank}A\le 1$, equivalently when all its $2\times 2$ minors vanish. Thusan ideal generated by three quadrics. By the stated assumption this is the full radical ideal of $Y$.

Every nonzero point of $Y$ has rank one. The action of $G{L}_3(k)\times G{L}_2(k)$ by row and column operations is transitive on rank-one matrices, so it suffices to compute atAt $A_0$, the differentials of the three minors are respectivelyHence $\dim T_{Y,A_0}=6-2=4$, and the same holds at every rank-one point. At the zero matrix all three quadrics have zero differential, so $T_{Y,0}={\mathbb{A}}^6$ has dimension six. It follows that

The identity theorem says that if two holomorphic functions on a connected Riemann surface agree on a set having an accumulation point, then they agree everywhere.

Apply local coordinates to their difference $F$. The power-series proof in the plane shows that near any point, either $F$ vanishes identically or its zeros are isolated: if the first nonzero Taylor coefficient has order $m$, then $F(z)=(z-z_0{)}^{m}G(z)$ with $G(z_0)\ne 0$. An accumulation point of the zero set therefore has a neighbourhood on which $F=0$. The set of points having such a neighbourhood is nonempty and open. It is also closed: near a limit point, a coordinate neighbourhood contains an open set on which $F=0$, and the planar identity theorem makes $F$ vanish throughout that coordinate neighbourhood. Connectedness now makes this set the whole surface.

For a nonempty connected open set $U\subset {\mathbb{R}}^2$, a function $h:U\to \mathbb{R}$ is harmonic when $h\in C^2(U)$ andOn a sufficiently small disc, the one-formis closed and hence equals $dk$ for some $k$. The Cauchy--Riemann equations then make $h+ik$ holomorphic. Holomorphic functions are smooth, so $h\in C^{\infty }(U)$.

A real-valued function $H$ on a Riemann surface is harmonic when, in every holomorphic chart $z$, its coordinate expression is harmonic. This definition is chart-independent. If $w=w(z)$ is a holomorphic transition map, the chain rule and Cauchy--Riemann equations giveThus vanishing of the Laplacian is preserved by every change of holomorphic coordinate.

A based loop is a continuous map $\alpha :[0,1]\to X$ with $\alpha (0)=\alpha (1)=x_0$. Two loops are equivalent when they are joined by a homotopy that keeps both endpoints at $x_0$. The fundamental group ${\pi }_1(X,x_0)$ is the set of these based-homotopy classes.

For loops $\alpha$ and $\beta$, define concatenation byIf $H$ and $K$ are based homotopies from $\alpha$ to ${\alpha }^{\prime }$ and from $\beta$ to ${\beta }^{\prime }$, concatenating $H(s,-)$ and $K(s,-)$ gives a based homotopy from $\alpha \ast \beta$ to ${\alpha }^{\prime }\ast {\beta }^{\prime }$. Thus multiplication $[\alpha ][\beta ]=[\alpha \ast \beta ]$ is well-defined.

Concatenating three paths with different breakpoints changes only the speed of traversal. Linear interpolation between the two increasing piecewise-linear parametrizations gives a based homotopy, proving associativity on classes. The constant loop $c(t)=x_0$ is an identity, since deleting its stationary half is another endpoint-fixing reparametrization. The inverse of $[\alpha ]$ is represented by $\bar{\alpha }(t)=\alpha (1-t)$. Indeed, $\alpha \ast \bar{\alpha }$ contracts throughand the analogous contraction handles $\bar{\alpha }\ast \alpha$. Hence the operation satisfies all group axioms.