## I.5: Nonsingular Varieties

### 5.1. #to-work

Locate the singular points and sketch the following curves in ${\mathbb{A}}^2$ (assume char $k \neq 2$ ). Which is which in Figure 4 ?

1.  $x^2=x^4+y^4$ :
2.  $x y=x^6+y^6$ :
3.  $x^3=y^2+x^4+y^4$ :
4.  $x^2 y+x y^2=x^4+y^4$.

![](attachments/2022-09-21_00-14-42.png)

### 5.2. #to-work

Locate the singular points and describe the singularities of the following surfaces in ${\mathbb{A}}^3$ (assume char $k \neq 2$ ). Which is which in Figure 5?

1.  $x y^2=z^2$
2.  $x^2+y^2=z^2$
3.  $xy + x^3 + y^3 = 0$.

![](attachments/2022-09-21_00-15-32.png)

### 5.3. Multiplicities. #to-work

Let $Y \subseteq {\mathbb{A}}^2$ be a curve defined by the equation $f(x, y)=0$. Let $P=(a, b)$ be a point of ${\mathbb{A}}^2$. Make a linear change of coordinates so that $P$ becomes the point $(0,0)$. Then write $f$ as a sum $f=f_0+f_1+\ldots+f_d$, where $f_i$ is a homogeneous polynomial of degree $i$ in $x$ and $y$. Then we define the multiplicity of $P$ on $Y$, denoted $\mu_P(Y)$, to be the least $r$ such that $f_r \neq 0$. (Note that $P \in Y \Leftrightarrow \mu_P(Y)>0$.) The linear factors of $f_r$ are called the tangent directions at $P$.

1.  Show that $\mu_P(Y) =1 \iff P$ is a nonsingular point of $Y$.
2.  Find the multiplicity of each of the singular points in (Ex. 5.1) above.

### 5.4. Intersection Multiplicity. #to-work

If $Y, Z \subseteq {\mathbb{A}}^2$ are two distinct curves, given by equations $f=0, g=0$, and if $P \in Y \cap Z$, we define the intersection multiplicity $(Y \cdot Z)_P$ of $Y$ and $Z$ at $P$ to be the length of the ${\mathcal{O}}_P$-module ${\mathcal{O}}_P /\left\langle{f, g}\right\rangle$.

1.  Show that $(Y \cdot Z)_P$ is finite, and $(Y \cdot Z)_P \geqslant \mu_P(Y) \cdot \mu_P(Z)$.
2.  If $P \in Y$, show that for almost all lines $L$ through $P$ (i.e., all but a finite number), $(L \cdot Y)_P=\mu_P(Y)$.
3.  If $Y$ is a curve of degree $d$ in ${\mathbb{P}}^2$, and if $L$ is a line in ${\mathbb{P}}^2, L \neq Y$, show that $(L \cdot Y)=d$. Here we define $(L \cdot Y)=\sum(L \cdot Y)_P$ taken over all points $P \in$ $L \cap Y$, where $(L \cdot Y)_p$ is defined using a suitable affine cover of ${\mathbb{P}}^2$.

### 5.5. #to-work

For every degree $d>0$, and every $p=0$ or a prime number, give the equation of a nonsingular curve of degree $d$ in ${\mathbb{P}}^2$ over a field $k$ of characteristic $p$.

### 5.6. Blowing Up Curve Singularities. #to-work

1.  Let $Y$ be the cusp or node of (Ex. 5.1). Show that the curve $\tilde{Y}$ obtained by blowing up $Y$ at $O=(0,0)$ is nonsingular (cf. (4.9.1) and (Ex. 4.10)).
2.  We define a node (also called ordinary double point) to be a double point (i.e., a point of multiplicity 2 ) of a plane curve with distinct tangent directions (Ex. 5.3). If $P$ is a node on a plane curve $Y$, show that $\varphi^{-1}(P)$ consists of two distinct nonsingular points on the blown-up curve $\tilde{Y}$. We say that "blowing up $P$ resolves the singularity at $P$".
3.  Let $P \in Y$ be the tacnode of $($ Ex. 5.1). If $\varphi: \tilde{Y} \rightarrow Y$ is the blowing-up at $P$. show that $\rho^{-1}(P)$ is a node. Using 2. we see that the tacnode can be resolved by two successive blowings-up.
4.  Let $Y$ be the plane curve $y^3=x^5$, which has a "higher order cusp" at $O$. Show that $O$ is a triple point: that blowing up $O$ gives rise to a double point (what kind?) and that one further blowing up resolves the singularity.

Note: We will see later $(\mathrm{V}, 3.8)$ that any singular point of a plane curve can be resolved by a finite sequence of successive blowings-up.

### 5.7. #to-work

Let $Y \subseteq {\mathbb{P}}^2$ be a nonsingular plane curve of degree $>1$, defined by the equation $f(x, y, z)=0$. Let $X \subseteq {\mathbb{A}}^3$ be the affine variety defined by $f$ (this is the cone over $Y$; see (Ex. 2.10) ). Let $P$ be the point $(0,0,0)$, which is the vertex of the cone. Let $\varphi: \tilde{X} \rightarrow X$ be the blowing-up of $X$ at $P$.

1.  Show that $X$ has just one singular point, namely $P$.
2.  Show that $\tilde{X}$ is nonsingular (cover it with open affines).
3.  Show that $\varphi^{-1}(P)$ is isomorphic to $Y$.

### 5.8. #to-work

Let $Y \subseteq {\mathbb{P}}^n$ be a projective variety of dimension $r$. Let $f_1, \ldots, f_t \in S=$ $k\left[x_0, \ldots, x_n\right]$ be homogeneous polynomials which generate the ideal of $Y$. Let $P \in Y$ be a point, with homogeneous coordinates $P=\left(a_0, \ldots, a_n\right)$. Show that $P$ is nonsingular on $Y$ if and only if the rank of the matrix $\left[ {\frac{\partial f_i}{\partial x_j}\,}(a_0,\cdots, a_n)\right]$ is $n-r$.[^1]

### 5.9. #to-work

Let $f \in k[x, y ; z]$ be a homogeneous polynomial, let $Y=Z(f) \subseteq {\mathbb{P}}^2$ be the algebraic set defined by $f$, and suppose that for every $P \in Y$, at least one of ${\frac{\partial f}{\partial x}\,}(P), {\frac{\partial f}{\partial y}\,}(P), {\frac{\partial f}{\partial z}\,}(P)$ is nonzero. Show that $f$ is irreducible (and hence that $Y$ is a nonsingular variety). [^2]

### 5.10. #to-work

For a point $P$ on a variety $X$. let ${\mathfrak{m}}$ be the maximal ideal of the local ring ${\mathcal{O}}_P$. We define the Zariski tangent space $T_P(X)$ of $X$ at $P$ to be the dual $k$-vector space of ${\mathfrak{m}}/{\mathfrak{m}}^2$.

1.  For any point $P \in X$. $\operatorname{dim} T_P(X) \geqslant \operatorname{dim} X$. with equality if and only if $P$ is nonsingular.
2.  For any morphism $\varphi: X \rightarrow Y$, there is a natural induced $k$-linear map $T_P(\varphi)$ : $T_P(X) \rightarrow T_{\varphi(P)}(Y)$
3.  If $\varphi$ is the vertical projection of the parabola $x=y^2$ onto the $x$-axis, show that the induced map $T_0(\varphi)$ of tangent spaces at the origin is the zero map.

### 5.11. The Elliptic Quartic Curve in ${\mathbb{P}}^3$. #to-work {#the-elliptic-quartic-curve-in-mathbbp3.-to-work}

Let $Y$ be the algebraic set in ${\mathbb{P}}^3$ defined by the equations $x^2-x z- yw=0$ and $yz -xw - zw = 0$. Let $P$ be the point $(x, y, z, w)=(0,0,0,1)$. and let $\varphi$ denote the projection from $P$ to the plane $w=0$. Show that $\varphi$ induces an isomorphism of $Y-P$ with the plane cubic curve $y^2 z-x^3+x z^2=0$ minus the point $(1,0,-1)$. Then show that $Y$ is an irreducible nonsingular curve. It is called the elliptic quartic curve in ${\mathbb{P}}^3$. Since it is defined by two equations it is another example of a complete intersection (Ex. 2.17).

### 5.12. Quadric Hypersurfaces. #to-work

Assume char $k \neq 2$. and let $f$ be a homogeneous polynomial of degree 2 in $x_0 \ldots \ldots x_n$.

1.  Show that after a suitable linear change of variables, $f$ can be brought into the form $f=x_0^2+\ldots+x_r^2$ for some $0 \leqslant r \leqslant n$.
2.  Show that $f$ is irreducible if and only if $r \geqslant 2$.
3.  Assume $r \geqslant 2$, and let $Q$ be the quadric hypersurface in ${\mathbb{P}}^n$ defined by $f$. Show that the singular locus $Z=\operatorname{Sing} Q$ of $Q$ is a linear variety (Ex. 2.11) of dimension $n-r-1$. In particular, $Q$ is nonsingular if and only if $r=n$.
4.  In case $r<n$, show that $Q$ is a cone with axis $Z$ over a nonsingular quadric hypersurface $Q^{\prime} \subseteq {\mathbb{P}}^r$.[^3]

### 5.13. #to-work

It is a fact that any regular local ring is an integrally closed domain (Matsumura $[2. \mathrm{Th}. 36, p. 121]$). Thus we see from (5.3) that any variety has a nonempty open subset of normal points (Ex. 3.17). In this exercise, show directly (without using (5.3)) that the set of nonnormal points of a variety is a proper closed subset (you will need the finiteness of integral closure: see (3.9A)).
  
### 5.14. Analytically Isomorphic Singularities. #to-work

1.  If $P \in Y$ and $Q \in Z$ are analytically isomorphic plane curve singularities, show that the multiplicities $\mu_P(Y)$ and $\mu_Q(Z)$ are the same (Ex. 5.3).
2.  Generalize the example in the text $(5.6 .3)$ to show that if $f=f_r+f_{r+1}+\ldots \in$ $k{\left[\left[ x, y \right]\right]  }$, and if the leading form $f_r$ of $f$ factors as $f_r=g_s h_t$, where $g_s, h_t$ are homogeneous of degrees $s$ and $t$ respectively, and have no common linear factor, then there are formal power series 
$$
\begin{align*}
g=g_s+g_{s+1}+\ldots \\
h=h_t+h_{t+1}+\ldots
\end{align*}
$$
    in $k{\left[\left[ x, y \right]\right]  }$ such that $f=gh$.
3.  Let $Y$ be defined by the equation $f(x, y)=0$ in ${\mathbb{A}}^2$, and let $P=(0,0)$ be a point of multiplicity $r$ on $Y$, so that when $f$ is expanded as a polynomial in $x$ and $y$, we have $f=f_r+$ higher terms. We say that $P$ is an ordinary $r{\hbox{-}}$fold point if $f_r$ is a product of $r$ distinct linear factors. Show that any two ordinary double points are analytically isomorphic. Ditto for ordinary triple points. But show that there is a one-parameter family of mutually nonisomorphic ordinary 4-fold points.
4.  \* Assume char $k \neq 2$. Show that any double point of a plane curve is analytically isomorphic to the singularity at $(0,0)$ of the curve $y^{2}=x^r$, for a uniquely determined $r \geqslant 2$. If $r=2$ it is a node (Ex. 5.6). If $r=3$ we call it a cusp: if $r=4$ a tacnode. See $(\mathrm{V}, 3.9 .5)$ for further discussion.

### 5.15. Families of Plane Curves. #to-work

A homogeneous polynomial $f$ of degree $d$ in three variables $x, y, z$ has ${d+2\choose 2}$ coefficients. Let these coefficients represent a point in ${\mathbb{P}}^N$. where $N= {d+2\choose 2} - 1 = {1\over 2}d(d+3)$.

1.  Show that this gives a correspondence between points of ${\mathbb{P}}^{N}$ and algebraic sets in ${\mathbb{P}}^2$ which can be defined by an equation of degree $d$. The correspondence is 1-1 except in some cases where $f$ has a multiple factor.
2.  Show under this correspondence that the (irreducible) nonsingular curves of degree $d$ correspond 1-1 to the points of a nonempty Zariski-open subset of ${\mathbb{P}}^N$. [^4]

[^1]: Hint: 1. Show that this rank is independent of the homogeneous coordinates chosen for $P$ : 2. pass to an open affine $U, \subseteq {\mathbb{P}}^n$ containing $P$ and use the affine Jacobian matrix: 3. you will need Euler's lemma, which says that if $f$ is a homogeneous polynomial of degree $d$, then $\sum x_i {\frac{\partial f}{\partial x_i}\,} = d\cdot f$.

[^2]: Hint: Use (Ex. 3.7).

[^3]: This notion of cone generalizes the one defined in (Ex. 2.10). If $Y$ is a closed subset of ${\mathbb{P}}^r$, and if $Z$ is a linear subspace of dimension $n-r-1$ in ${\mathbb{P}}^n$, we embed ${\mathbb{P}}^r$ in ${\mathbb{P}}^n$ so that ${\mathbb{P}}^r \cap Z=\varnothing$, and define the cone over $Y$ with axis $Z$ to be the union of all lines joining a point of $Y$ to a point of $Z$.

[^4]: Hints: (1) Use elimination theory (5.7A) applied to the homogeneous polynomials ${\frac{\partial f}{\partial x_0}\,}, \cdots, {\frac{\partial f}{\partial x_n}\,}$ use the previous (Ex. 5.5, 5.8, 5.9) above.