Blowups, Exceptional Divisors, and Strict Transforms: Examples and Counterexamples

The blowup is $V(xv-yu)\subseteq {\mathbb{A}}_k^2{\times }_k{\mathbb{P}}_k^1$ by [F1], and its two standard charts are $\mathrm{Spec}k[x,T]$ with $y=xT$ and $\mathrm{Spec}k[y,U]$ with $x=yU$; these are exactly the affine blowup algebras $k[x,y][I\mathrm{/}x]$ and $k[x,y][I\mathrm{/}y]$ of [F2] and [F3], and by [F5] the two charts of ${\mathbb{P}}_k^1$ glue along $TU=1$.

Each chart ring is a polynomial ring in two variables over $k$, namely $k[x,T]\cong k[x,y\mathrm{/}x]$ and $k[y,U]\cong k[x\mathrm{/}y,y]$, so $\mathrm{Spec}k[x,y][I\mathrm{/}x]=\mathrm{Spec}k[x,T]\cong {\mathbb{A}}_k^2$ and $\mathrm{Spec}k[y,U]\cong {\mathbb{A}}_k^2$; the overlap is the open subscheme $D(T)=D(U^{-1})$ of the first chart, identified with $D(U)$ in the second by $T\mapsto U^{-1}$.

By [F1] the exceptional divisor is cut by $x$ in the first chart and by $y$ in the second, and it is ${\mathbb{P}}_k^1$: in the first chart $E\cap \{x\mathrm{\ne }0\}$ is empty and $E$ is the line $V(x)\cong \mathrm{Spec}k[T]$, in the second $E=V(y)\cong \mathrm{Spec}k[U]$, and the two affine lines glue along $TU=1$ by [F5].

The projection sends the first chart to ${\mathbb{A}}_k^2$ by $(x,T)\mapsto (x,xT)$ and the second by $(y,U)\mapsto (yU,y)$: on the open locus $x\mathrm{\ne }0$ of the first chart the formula is inverted by $T=y\mathrm{/}x$, so the projection restricts to an isomorphism onto $\{(x,y):x\mathrm{\ne }0\}$, and symmetrically the second chart is isomorphic to $\{(x,y):y\mathrm{\ne }0\}$ over the base. These two open subschemes cover ${\mathbb{A}}_k^2\setminus \{0\}$ and the inverses agree on the overlap because $T=U^{-1}$ there (step 2.1), so the projection is the identity over ${\mathbb{A}}_k^2\setminus \{0\}$. It contracts $E$ to the origin: on $E$ the first chart has $x=0$ and image $(0,0)$, and on the second $y=0$ and image $(0,0)$, while every point of $E$ lies in one of the two charts (step 3.1).

Let $L=V(\mathrm{\ell })$ be a line through the origin and first suppose $\mathrm{\ell }=y-ax$ with $a\in k$. In the first chart the pulled-back equation is $xT-ax=x(T-a)$, so the pullback divisor is the sum of $E=V(x)$ and the strict transform $V(T-a)$; in the second chart it is $y-ayU=y(1-aU)$, the sum of $E=V(y)$ and $V(1-aU)$, and for $a\mathrm{\ne }0$ the two strict-transform pieces glue at the same exceptional point with coordinates $T=a$, $U=a^{-1}$; for $a=0$ the second piece is empty and the exceptional intersection is $T=0$. For the vertical line $\mathrm{\ell }=x$ the first chart gives $x$, namely $E$, and the second gives $yU$, namely $E$ plus the strict transform $V(U)$; so in both cases the total transform is the strict transform plus $E$, with multiplicity one along $E$.

The three charts of the blowup are $\mathrm{Spec}A[I\mathrm{/}x]$, $\mathrm{Spec}A[I\mathrm{/}y]$, $\mathrm{Spec}A[I\mathrm{/}z]$ by [F1]; by [F2] the first is $A$-generated by $y\mathrm{/}x$ and $z\mathrm{/}x$, with $IA[I\mathrm{/}x]=xA[I\mathrm{/}x]$, the map from $k[x,s,t]$ sending $s$ to $y\mathrm{/}x$ and $t$ to $z\mathrm{/}x$ is injective, since after inverting $x$ it is the coordinate change $y=xs$, $z=xt$ in $k[x,x^{-1},y,z]$. Thus it is the polynomial ring $k[x,y\mathrm{/}x,z\mathrm{/}x]\cong k[x,s,t]$, whose spectrum is ${\mathbb{A}}_k^3$; symmetrically the other two charts are $k[x\mathrm{/}y,y,z\mathrm{/}y]$ and $k[x\mathrm{/}z,y\mathrm{/}z,z]$, each isomorphic to ${\mathbb{A}}_k^3$, glued by the transition ratios of [F1] and [F3].

The same blowup is presented inside ${\mathbb{A}}^3\times {\mathbb{P}}^2$ by the $2\times 2$ minors of $\left(\begin{array}{ccc}x& y& z\\ T_0& T_1& T_2\end{array}\right)$: on the chart $D_{+}(T_0)$ set $T_0=1$. The minor equations become $x{T}_1=y$, $x{T}_2=z$, the third minor being a consequence of these two. Eliminating $y,z$ gives exactly the first polynomial chart of step 1.1. The other charts are symmetric; their ratio transitions coincide with those of the blowup, so the chart isomorphisms glue to the claimed closed incidence subscheme, without needing a separate assertion about the kernel of the entire Rees presentation.

The exceptional divisor is cut by $x$ in the first chart, by $y$ in the second and by $z$ in the third: on each chart $I\mathcal{O}$ is generated by the corresponding variable, by [F2] and [F4]. By [F5] and [F6] it is ${\mathrm{Proj}}_k({\mathrm{gr}}_{(x,y,z)}k[x,y,z{]}_0)={\mathrm{Proj}}_{k}k[X,Y,Z]={\mathbb{P}}_k^2$, the projectivized cotangent space in the quotient convention, since the conormal space $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ is free of rank three over $k$.

Finally [F7] gives ${\mathcal{O}}_{\mathrm{Bl}}(E)=\mathcal{O}(-1)$ and ${\mathcal{O}}_{\mathrm{Bl}}(-E)=I{\mathcal{O}}_{\mathrm{Bl}}=\mathcal{O}(1)$; restricting the first identity to $E$ and using the identification $E={\mathrm{Proj}}_k{\mathrm{gr}}_{\mathfrak{m}}k[x,y,z{]}_0={\mathbb{P}}_k^2$ of step 3.1, the restricted twist is the standard ${\mathcal{O}}_{{\mathbb{P}}_k^2}(-1)$, so ${\mathcal{O}}_E(E)={\mathcal{O}}_{{\mathbb{P}}_k^2}(-1)$ in the quotient convention.

In the notation of the given data, multiplication by $f$ defines an $A$-module map $A\to I$ that is surjective because $I=(f)$, and injective because $f$ is a nonzerodivisor; it is therefore an isomorphism, so $I$ is a free $A$-module of rank one and the ideal sheaf $\tilde{I}$ is invertible. The same nonzerodivisor condition says that $f$, read as the global local equation of $D=V(I)$ on $\mathrm{Spec}A$, is a regular section, so $D$ is an effective Cartier divisor with ideal sheaf ${\mathcal{I}}_D=\tilde{I}$.

By step 1.1 the ideal sheaf $\tilde{I}$ is invertible, equivalently $D$ is an effective Cartier divisor with that ideal sheaf, so [F3] applies to the blowup of $\mathrm{Spec}A$ along $I$ and shows that $\pi \text{\hspace{0.17em}⁣}:{\mathrm{Bl}}_I\mathrm{Spec}A\to \mathrm{Spec}A$ is an isomorphism.

Independently of step 2.1, compute the chart: since $I=(f)$, the Rees algebra is $R(I)={⨁}_{n\ge 0}(f{)}^n{t}^n=A[ft]$, and the generating family $(f)$ has the single element $f$, so the standard chart $D_{+}(ft)$ of the blowup is $\mathrm{Spec}{B}_0$ with $B_0=A[I\mathrm{/}f]=(A[ft][(ft{)}^{-1}]{)}_0$. An element of $A[ft][(ft{)}^{-1}]=A[ft,(ft{)}^{-1}]$ is a finite sum ${\sum }_{k\in \mathbb{Z}}{a}_k(ft{)}^k$ with $a_k\in A$; it has degree zero exactly when $a_k=0$ for every $k\mathrm{\ne }0$, so $B_0=A$ and $A[(f)\mathrm{/}f]=A$. The chart $D_{+}(ft)$ covers the whole blowup, so there are no other charts to compare.

Steps 2.1 and 3.1 agree: the blowup is $\mathrm{Spec}A$ with identity structural morphism, and its single affine blowup algebra is $A[(f)\mathrm{/}f]=A$. The geometric instances named in the statement are covered by the same computation: the ideal of a $k$-rational point of a regular curve is generated at that point by a uniformizer, hence by a nonzerodivisor equation of an effective Cartier divisor, and the ideal of a line in the plane is generated by a linear form, again a nonzerodivisor; blowing up either changes nothing.

In the special case $d=2$ of [F1], the Rees algebra of $I^2$ has degree-$n$ piece $I^{2n}$, which by [F2] is exactly the degree-$2n$ piece of $\mathcal{R}(I)$; hence $\mathcal{R}(I^2)=\mathcal{R}(I{)}^{(2)}$ as graded algebras, and the canonical identification of Proj's from [F3] (with its identity on coordinate rings $S_{(f)}=S_{(f^2)}^{(2)}$) glues by [F1] to a canonical isomorphism ${\mathrm{Bl}}_{I^2}{A}^2\to {\mathrm{Bl}}_I{A}^2$ over $A^2$.

Concretely, [F4] presents ${\mathrm{Bl}}_I{A}^2$ by the two charts $A[x,y][I\mathrm{/}x]=A[x,y\mathrm{/}x]$ and $A[x,y][I\mathrm{/}y]=A[x\mathrm{/}y,y]$, glued by inverting the ratio; in the first chart put $T=y\mathrm{/}x$, so the chart is $A[x,T]$ and its exceptional divisor is $V(x)$. The chart of the same kind for $I^2$ is $A[x,y][I^2\mathrm{/}{x}^2]$, and the degree-two generators $x^2,xy,y^2$ give the fractions $xy\mathrm{/}{x}^2=y\mathrm{/}x=T$ and $y^2\mathrm{/}{x}^2=T^2$, so this chart ring is $A[x,T]$ again; symmetrically the second charts agree as subrings of $A[x,x^{-1},y]$ and $A[y,y^{-1},x]$. Writing $X,Y,Z$ for their degree-one Rees symbols, their relations include $XZ=Y^2$, $xY=yX$ and $xZ=yY$; the Veronese identification in step 1.1 gives the same blowup. The two charts cover also the regraded Proj, since $XZ=Y^2$ prevents a homogeneous prime outside the irrelevant locus from containing both $X$ and $Z$. Its original incidence description is checked directly: intersecting $V(x{T}_1-y{T}_0)\subseteq A^2\times {\mathbb{P}}^1$ with the chart $T_0=1$ gives $xT=y$ with $T=T_1\mathrm{/}{T}_0$, namely the first chart, and with $T_1=1$ gives $x=yU$, $U=T_0\mathrm{/}{T}_1$, namely the second, the two glued by $TU=1$.

Thus the identity morphism on the underlying charts, read through the Veronese regrading of step 1.1, is the canonical isomorphism ${\mathrm{Bl}}_I{A}^2\cong {\mathrm{Bl}}_{I^2}{A}^2$, and the second description uses the degree-two generators $x^2,xy,y^2$ with their relation $XZ=Y^2$ (the degree-two Veronese conic in ${\mathbb{P}}^2$) as claimed.

In the first chart of [F2] write $s=T=y\mathrm{/}x$, so the chart ring is $k[x,s]$ with $y=xs$ and $E=V(x)$, and let $f=y^2-x^3$, a reduced equation of the cusp of multiplicity $m=2$ at the origin; substituting $y=xs$ gives $f(x,xs)=x^2{s}^2-x^3=x^2(s^2-x)$ with $s^2-x$ not divisible by $x$ because its reduction modulo $x$ is $s^2\mathrm{\ne }0$, so by [F1] (or [F4]) the strict transform is cut in this chart by $g=s^2-x$, and by [F3] the strict transform is the closure of the corresponding open part.

The curve $g=s^2-x=0$ is regular: its gradient $(-1,2s)$ never vanishes, so $C^{\mathrm{\prime }}$ is the smooth parabola $x=s^2$; its intersection with $E=V(x)$ in this chart is $V(x,s^2-x)=V(x,s^2)$, the single point $s=0$, and the local ring of $C^{\mathrm{\prime }}$ there is $k[s{]}_{(s)}$ with the equation of $E$ restricting to $s^2$, so the contact order is $\mathrm{length}(k[s{]}_{(s)}\mathrm{/}(s^2))=2$; both $C^{\mathrm{\prime }}$ and $E$ are regular at this point with the same tangent line, so the curves are tangent there. This agrees with [F5]: the leading form of $f$ is $f_2=y^2$, whose dehomogenization $f_2(1,s)=s^2$ vanishes only at $s=0$, with multiplicity $2$, so $C^{\mathrm{\prime }}\cap E$ is the single point of multiplicity $2$.

In the second chart of [F2] write $U=x\mathrm{/}y$, so the chart ring is $k[y,U]$ with $x=yU$ and $E=V(y)$; substituting gives $f=y^2-y^3{U}^3=y^2(1-y{U}^3)$, and the residual factor $1-y{U}^3$ is a unit at every point of $E$ (where $y=0$ it equals $1$), so the strict transform has no points of $E$ in this chart.

The total transform identity ${\pi }^{\ast }C=C^{\mathrm{\prime }}+2E$ of [F4] is visible in the two charts of steps 1.1 and 3.1: the pullback of $y^2-x^3$ factors as $x^2(s^2-x)$ and as $y^2(1-y{U}^3)$, the exceptional factor $x^2$ or $y^2$ contributing $2E$ and the residual factor the strict transform; no other component of $E$ appears, since the residual factors do not vanish along $E$.

The point $V(x,s)$ at which $C^{\mathrm{\prime }}$ is tangent to $E$ is a point at which the two distinct regular curves $C^{\mathrm{\prime }}$ and $E$ meet with contact order $n=2$; blowing up this point, [F6] applies with $n>1$ and shows that the strict transforms of $C^{\mathrm{\prime }}$ and of $E$ meet, at the point of the new exceptional curve corresponding to their common tangent direction, with contact order $n-1=1$, that is, transversally: the tangency is separated by the second blowup.

Therefore in the first chart the strict transform is the smooth parabola $s^2=x$, meeting $E=(x=0)$ at the single point $s=0$ with multiplicity $2$, while in the second chart the strict transform does not meet $E$; the cusp has become a regular curve tangent to $E$, and the second point blowup reduces the contact order of $C^{\mathrm{\prime }}$ with $E$ to one, separating the tangency.

In the first chart of [F2] write $s=T=y\mathrm{/}x$, so the chart ring is $k[x,s]$ with $y=xs$ and $E=V(x)$, and let $f=y^2-x^3-x^2$, a reduced equation of $C$ of multiplicity $m=2$ at the origin with leading form $f_2=y^2-x^2=(y-x)(y+x)$, which is squarefree because the characteristic is not $2$; substituting $y=xs$ gives $f(x,xs)=x^2{s}^2-x^3-x^2=x^2(s^2-x-1)$ with $s^2-x-1$ not divisible by $x$, since its reduction modulo $x$ is $s^2-1$, so the strict transform is cut in this chart by $g=s^2-x-1$ by [F1] and [F3].

In this chart $C^{\mathrm{\prime }}\cap E=V(x,s^2-x-1)=V(x,s^2-1)$, the two distinct points $s=1$ and $s=-1$; at each of them the local ring of $C^{\mathrm{\prime }}$ is $k[s{]}_{(s\mp 1)}$ with the equation of $E$ restricting to $s^2-1=(s-1)(s+1)$, which has a simple zero at each point, so the contact order is one and $C^{\mathrm{\prime }}$ meets $E$ transversally there; this agrees with [F4], since $f_2(1,s)=s^2-1=(s-1)(s+1)$ is squarefree with the two distinct roots $s=\pm 1$, the two tangent directions of $C$ at the origin. The curve $C^{\mathrm{\prime }}=V(s^2-x-1)$ is regular, its gradient $(-1,2s)$ being nowhere zero, so the strict transform is regular at both points.

In the second chart of [F2] write $U=x\mathrm{/}y$, so the chart ring is $k[y,U]$ with $x=yU$ and $E=V(y)$; substituting gives $f=y^2-y^3{U}^3-y^2{U}^2=y^2(1-U^2-y{U}^3)$, so the strict transform is cut in this chart by $1-U^2-y{U}^3$ and meets $E$ where $y=0$ and $1-U^2=0$, namely at $U=1$ and $U=-1$; these are the same two points as $s=1$ and $s=-1$, because $U=1\mathrm{/}s$ on the overlap $TU=1$ of [F2], and there is no further point of $E$ on $C^{\mathrm{\prime }}$ in this chart, so the other chart contributes no additional points of $E$.

Consequently $C^{\mathrm{\prime }}\cap E$ consists exactly of the two distinct points over the node, one for each of the two factors $y-x$ and $y+x$ of the leading form, so the two branches of the node, whose tangent directions are those two factors, arrive at distinct points of $E$ and are separated by the one point blowup; the strict transform $C^{\mathrm{\prime }}$ is regular and transverse to $E$ at both points, and in the first chart it is the smooth conic-like curve $V(s^2-x-1)$ while in the second chart it is $V(1-U^2-y{U}^3)$, the two descriptions agreeing on the overlap.