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 ?
- \(x^2=x^4+y^4\) :
- \(x y=x^6+y^6\) :
- \(x^3=y^2+x^4+y^4\) :
- \(x^2 y+x y^2=x^4+y^4\).
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?
- \(x y^2=z^2\)
- \(x^2+y^2=z^2\)
- \(xy + x^3 + y^3 = 0\).
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 \(P\).
- Show that \(\mu_P(Y) =1 \iff P\) is a nonsingular point of \(Y\).
- 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 \({\mathcal{O}}_P /\left\langle{f, g}\right\rangle\).
- Show that \((Y \cdot Z)_P\) is finite, and \((Y \cdot Z)_P \geqslant \mu_P(Y) \cdot \mu_P(Z)\).
- If \(P \in Y\), show that for almost all lines \(L\) through \(P\) (i.e., all but a finite number), \((L \cdot Y)_P=\mu_P(Y)\).
- 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
- 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)).
- 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 \(P\)”.
- 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.
- 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 \(P\).
- Show that \(X\) has just one singular point, namely \(P\).
- Show that \(\tilde{X}\) is nonsingular (cover it with open affines).
- 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 \({\mathfrak{m}}/{\mathfrak{m}}^2\).
- For any point \(P \in X\). \(\operatorname{dim} T_P(X) \geqslant \operatorname{dim} X\). with equality if and only if \(P\) is nonsingular.
- For any morphism \(\varphi: X \rightarrow Y\), there is a natural induced \(k\)-linear map \(T_P(\varphi)\) : \(T_P(X) \rightarrow T_{\varphi(P)}(Y)\)
- 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
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 \(x_0 \ldots \ldots x_n\).
- 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 \(0 \leqslant r \leqslant n\).
- Show that \(f\) is irreducible if and only if \(r \geqslant 2\).
- 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 \(r=n\).
- 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
- 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).
- 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*} \begin{align*} g=g_s+g_{s+1}+\ldots \\ h=h_t+h_{t+1}+\ldots \end{align*} \end{align*} in \(k{\left[\left[ x, y \right]\right] }\) such that \(f=gh\).
- 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.
- * 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 \(N= {d+2\choose 2} - 1 = {1\over 2}d(d+3)\).
- 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.
- 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