Smooth and singular points; Jacobian and regularity criteria (Tag D-0SYCY)
For \(p \in X\) with local ring \(\OO_{X,p}\) and maximal ideal \(\mfm_p\), the Zariski tangent space is \[ T_p X \da \dualof{(\mfm_p / \mfm_p^2)} . \] The point \(p\) is smooth if \(\dim_k T_p X = \dim_p X\), and singular otherwise.
(Jacobian criterion)
Let \(X\subseteq\AA^n\) be an affine variety of dimension \(d\), let \(I(X) = \gens{f_1,\ldots,f_t} \subseteq k[x_1,\ldots,x_n]\), and let \(J_X(p) = \left[ \partial f_i / \partial x_j (p) \right]\). Then \(\rank J_X(p)\le n-d\) for every \(p\in X\), and \(p\) is a smooth point if and only if \(\rank J_X(p) = n-d\) [@Har10a, Theorem I.5.1].
For every \(p\in X\), \(\dim_k T_pX\ge\dim_pX\). The point \(p\) is smooth if and only if \(\OO_{X,p}\) is a regular local ring, and regularity of the local ring is defined at every point of a scheme, closed or not.
The \(f_i\) in the Jacobian criterion generate the ideal \(I(X)\). Equations that cut out \(X\) only as a set do not suffice: \(V(x^2)=\{0\}\subseteq\AA^1\) has \(J(0)=(0)\), of rank \(0\ne1\), although \(0\) is a smooth point of \(\{0\}=V(x)\).
Over a perfect field, the Jacobian condition at \(p\) is equivalent to regularity of \(\OO_{X,p}\). Over an imperfect field, the Jacobian condition is smoothness, which is preserved by base change to \(\bar{k}\); regularity of \(\OO_{X,p}\) is not. Over \(k = \FF_p(t)\) the closed subscheme \(V(x^p - t) \subseteq \AA^1\) is the spectrum of a field, hence regular, but it is not smooth: after base change to \(k(t^{1/p})\) it becomes \(V((x - t^{1/p})^p)\), which is not reduced.
The Jacobian rank condition is computational for a presented variety, while regularity of the local ring is intrinsic and extends to schemes. Over a perfect field the two criteria agree.
Smooth hyperplane sections
Bertini's theorem (Tag T-BERTINI)
Let \(X \subseteq \PP^n\) be a smooth projective variety over an algebraically closed field \(k\) of any characteristic. The hyperplanes \(H \subseteq \PP^n\) with \(H \not\supseteq X\) and \(H \cap X\) smooth form a dense open subset of \(\dualof{(\PP^n)}\). If \(X\) is irreducible of dimension at least \(2\), then \(H \cap X\) is moreover connected, hence irreducible, for \(H\) in that open set.
(Bertini's theorem for linear systems)
Let \(X\) be a smooth variety over an algebraically closed field of characteristic zero and let \(\mathfrak{d}\) be a linear system on \(X\). Then a general member of \(\mathfrak{d}\) is smooth away from the base locus of \(\mathfrak{d}\).
The second statement fails in characteristic \(p\). On \(\AA^2\) the pencil spanned by \(x^p\) and \(y^p\) has base locus the origin, and every member \(a x^p + b y^p = (a^{1/p} x + b^{1/p} y)^p\) is a nonreduced line, singular at each of its points.
Normal crossings
Normal crossing singularities (Tag D-VARNCROSS)
Let \(k\) be an algebraically closed field. A variety \(Y\) of dimension \(n\) has a normal crossing singularity at a closed point \(p\) if there is an isomorphism of complete local rings \[ \hat{\OO}_{Y,p} \cong k[[x_0, \ldots, x_n]] / (x_0 x_1 \cdots x_r) \] for some \(0 \leq r \leq n\). That is, the completion of \(Y\) at \(p\) is isomorphic to the completion at the origin of the union of the \(r+1\) coordinate hyperplanes \(V(x_0),\ldots,V(x_r)\) in \(\AA^{n+1}\). \(Y\) has normal crossings if it has a normal crossing singularity or a smooth point at every closed point; the case \(r = 0\) is a smooth point.
(Simple normal crossing divisor)
Let \(X\) be a smooth variety and \(D = \sum_i D_i\) a reduced effective divisor. \(D\) is a normal crossing divisor if at each closed point \(p \in D\) there are local parameters \(x_1, \ldots, x_n\) of \(\hat{\OO}_{X,p}\) with \(D\) defined by \(x_1 \cdots x_r = 0\). \(D\) is a simple normal crossing divisor if moreover every component \(D_i\) is smooth, and at each point the components through \(p\) are cut out by distinct members of a single system of local parameters in \(\OO_{X,p}\) itself.
In characteristic not \(2\), the node \(y^2 = x^2(x+1)\) in \(\AA^2\) has a normal crossing singularity at the origin: \(x\sqrt{x+1}\) is a power series, and \(y^2 - x^2(x+1) = (y - x\sqrt{x+1})(y + x\sqrt{x+1})\) is a product of two parameters. As a divisor on \(\AA^2\) it is normal crossing but not simple normal crossing, since its one component is singular. The cusp \(y^2 = x^3\) is not a normal crossing singularity, since its tangent cone \(y^2 = 0\) is not a pair of distinct lines.
Tangent hyperplanes and the dual variety
The dual variety of a projective variety (Tag D-VARDUAL)
Let \(X \subseteq \PP^n\) be an irreducible projective variety, and let \(\dualof{(\PP^n)}\) be the projective space of hyperplanes in \(\PP^n\). A hyperplane \(H\) is tangent to \(X\) at a smooth point \(p\) when \(H\) contains the projective tangent space \(\mathbb{T}_p X\). The dual variety \(\dualof{X} \subseteq \dualof{(\PP^n)}\) is the closure of the set of hyperplanes tangent to \(X\) at some smooth point.
The incidence correspondence \[ \Phi = \overline{\ts{ (p, H) \st p \in X_{\mathrm{sm}},\ \mathbb{T}_p X \subseteq H }} \subseteq X \times \dualof{(\PP^n)} \] is irreducible of dimension \(n-1\), since over each smooth point \(p\) the fibre is a projective space of dimension \(n - 1 - \dim X\). So \(\dualof{X}\), the image of \(\Phi\) under the second projection, is irreducible of dimension at most \(n-1\). Over a field of characteristic zero, \(\dualof{(\dualof{X})} = X\).
If \(X \subseteq \PP^2\) is a smooth conic \(x^\top A x = 0\) with \(A\) symmetric and invertible, the tangent line at \(p\) has coordinates \(Ap\), and \(\dualof{X}\) is the conic \(\xi^\top A^{-1} \xi = 0\). A linear subspace \(\PP^k \subseteq \PP^n\) has dual variety the linear subspace of hyperplanes containing it, of dimension \(n-k-1\), so a dual variety need not be a hypersurface.
Normality, the weaker condition
Normal domains and schemes, and their relation to regularity (Tag D-QJ5M9)
An integral domain is normal if it is integrally closed in its fraction field. A scheme \(X\) is normal if every local ring \(\OO_{X,p}\), for \(p \in X\), is a normal domain.
A regular local ring is normal, and the converse fails in dimension \(\geq 2\). In dimension \(1\) the two agree: a Noetherian local domain of dimension \(1\) is normal exactly when it is a discrete valuation ring, that is, regular.
(Normalization)
Let \(X\) be an integral scheme. A normalization of \(X\) is a normal integral scheme \(\widetilde{X}\) with a dominant morphism \(\nu \colon \widetilde{X} \to X\) such that every dominant morphism \(f \colon Y \to X\) from a normal integral scheme \(Y\) factors uniquely as \(f = \nu \circ \tilde{f}\) for a morphism \(\tilde{f} \colon Y \to \widetilde{X}\).
(Existence of the normalization)
Every integral scheme \(X\) has a normalization, unique up to unique isomorphism. If \(X = \Spec A\) is affine, \(\widetilde{X} = \Spec \widetilde{A}\) for the integral closure \(\widetilde{A}\) of \(A\) in its fraction field, and in general \(\widetilde{X}\) is obtained by gluing these over an affine cover. If \(X\) is a variety over a field \(k\), then \(\nu\) is finite and birational, and \(\widetilde{X}\) is projective when \(X\) is projective [@Har10a, Exercise II.3.8].
The normalization of an integral curve is regular, hence smooth over a perfect field. Thus normalization resolves singularities of curves over perfect fields.
The quadric cone \(V(xy - z^2) \subseteq \AA^3\) is normal and singular at the origin: it is regular in codimension one and Cohen–Macaulay, so it is normal by Serre’s criterion \(R_1 + S_2\). The cuspidal cubic, with coordinate ring \(k[t^2,t^3] \subseteq k[t]\), is not normal, and its normalization is \(\Spec k[t]\).
A normal variety is regular in codimension one, so its singular locus has codimension at least two. A normal curve is therefore smooth: normalization resolves curve singularities, and the local rings at closed points of a smooth curve are discrete valuation rings.
Where the singular points are
Finding them is the Jacobian computation: set the partials to zero, intersect with the variety, and use the Euler relation in \(\PP^n\): \[ \sum_i x_i \frac{\partial f}{\partial x_i} = (\deg f) \cdot f \] makes the vanishing of the partials imply the vanishing of \(f\) whenever \(\deg f\) is invertible in \(k\). So for a plane curve in characteristic zero the singular locus is cut out by the partials alone.
Resolutions
Log resolutions (Tag D-VARLOGRES)
Let \(X\) be a variety and \(D \subseteq X\) a closed subset of codimension at least \(1\), possibly empty. A log resolution of the pair \((X, D)\) is a proper birational morphism \(f \colon Y \to X\) such that \(Y\) is smooth, the exceptional locus \(\operatorname{Exc}(f)\) is a divisor, and \(\operatorname{Exc}(f) \cup f^{-1}(D)\) is a simple normal crossing divisor.
(Hironaka)
Over a field of characteristic \(0\), every pair \((X, D)\) has a log resolution, which can be taken to be a composite of blowups along smooth centres and an isomorphism over the open set where \(X\) is smooth and \(D\) is a simple normal crossing divisor.
For the cuspidal curve \(D = V(y^2 - x^3) \subseteq \AA^2 = X\), one blowup at the origin makes the strict transform smooth but tangent to the exceptional curve \(E_1\). A second blowup leaves the strict transform and the two exceptional curves passing through one point, which is not a normal crossing. A third blowup at that point gives a log resolution: the total transform is a simple normal crossing divisor.
Crepant morphisms and crepant resolutions (Tag D-VARCREPANT)
Let \(X\) and \(Y\) be normal varieties whose canonical divisors are \(\QQ\)-Cartier. A proper birational morphism \(f \colon Y \to X\) is crepant if \(K_Y = f^* K_X\) as \(\QQ\)-divisor classes, that is, every \(f\)-exceptional prime divisor has discrepancy \(0\). A crepant resolution is a crepant \(f\) with \(Y\) smooth.
The minimal resolution of an ADE surface singularity is crepant: its exceptional curves \(E\) are smooth rational curves with \(E^2 = -2\), so \(K_Y \cdot E = 0\) for each, and \(K_Y - f^* K_X\), supported on the exceptional curves, is zero because the intersection form on them is negative definite. The blowup \(f \colon Y \to X\) of a smooth point on a surface is not crepant: \(K_Y = f^* K_X + E\). The quotient \(\CC^4 / \{\pm 1\}\) has no crepant resolution, because its singularity is terminal: every exceptional divisor of every resolution has positive discrepancy.
ADE surface singularities (Tag D-SRFADE)
An ADE singularity, also called a du Val singularity or rational double point, is a surface singularity over \(\CC\) analytically isomorphic to the singularity at the origin of one of the following hypersurfaces in \(\AA^3\):
| Type | Equation | Group \(G \subseteq \SL_2(\CC)\) |
|---|---|---|
| \(A_n\), \(n \geq 1\) | \(x^2 + y^2 + z^{n+1} = 0\) | cyclic of order \(n+1\) |
| \(D_n\), \(n \geq 4\) | \(x^2 + y^2 z + z^{n-1} = 0\) | binary dihedral of order \(4(n-2)\) |
| \(E_6\) | \(x^2 + y^3 + z^4 = 0\) | binary tetrahedral of order \(24\) |
| \(E_7\) | \(x^2 + y^3 + y z^3 = 0\) | binary octahedral of order \(48\) |
| \(E_8\) | \(x^2 + y^3 + z^5 = 0\) | binary icosahedral of order \(120\) |
The ADE singularities are exactly the quotient singularities \(\CC^2 / G\) for finite subgroups \(G \subseteq \SL_2(\CC)\), with \(G\) as in the table.
They are exactly the rational double points, and exactly the canonical surface singularities.
The exceptional locus of the minimal resolution is a union of smooth rational curves of self-intersection \(-2\) whose dual graph is the Dynkin diagram of the same name, and the minimal resolution is crepant.
\(A_1\) is the quadric cone \(x^2 + y^2 + z^2 = 0\), the quotient \(\CC^2 / \{\pm 1\}\); its minimal resolution is the blowup of the vertex, with one \((-2)\)-curve.