Deformation to the Normal Cone

An outline of blowup

Blow-up algebra (Rees algebra)

Given a ring \(R\) with ideal \(I\subset R\), we define the blow-up algebra (Rees algebra) for \(R\) at \(I\) to be the \(R\)-graded algebra

\[ Bl_I(R)=R\oplus I\oplus I^2\oplus\cdots \]

where the summand \(I^n\) is in degree \(n\). Let \(a\in I\) a (non-zero) element. Denote \(a^{(1)}\) the exact element \(a\) viewed as an element in degree \(1\) in \(Bl_I(R)\). Then we define the affine blowup algebra \(R[\frac{I}{a}]\) to be the degree \(0\) part of \(Bl_I(R)[(a^{(1)})^{-1}]\). Namely, this is the ring of elements in the form of \(x/a^n\) where \(x\in I^n\). Two representatives \(x/a^n\) and \(y/a^m\) are equal if and only if \(a^k(a^mx-a^ny)=0\) for some \(k\geq 0\). There is a canonical morphism from \(R\) to \(R[\frac{I}{a}]\) in the trivial way, sending \(a\in R\) to the element at degree \(0\).

Following the notation, we have

• The image of \(a\) in \(R[\frac{I}{a}]\) is a non-zerodivisor.
• \(IR[\frac{I}{a}]=aR[\frac{I}{a}]\).
• \(R\) and \(R[\frac{I}{a}]\) are isomorphic on local \(D(a)\).

Proof. If \(a\cdot(x/a^n)=0\) for some \(x\in I^n\), we have \(ax/(a^{n+1})=0\). Furthermore, \(a^n\cdot(x/a^n)=x\) so \(a\) generates \(I\) in \(R[\frac{I}{a}]\).

Blow-up algebra as local structure of blowups

For a scheme \(X\) together with an ideal sheaf \(\mathcal{I}\) corresponding to a closed subscheme \(Z\subset X\) we define the blowup of \(X\) along the center \(Z\) to be

\[ b:Proj_X(\oplus_{n\geq 0}\mathcal{I}^n)\rightarrow X. \]

The process of blowing up a scheme, in abstract sense, is to turn a chosen closed local into a divisor, that is, a locally principle ideal which is also an invertible sheaf. We can see this from the definition.

0804

Following the preceding notations, let \(U=spec(A)\) be an affine open subscheme of \(X\) and \(I\subset A\) corresponding to \(\mathcal{I}_U\). If \(b:X'\rightarrow X\) is the blowup along closed local cut out by \(\mathcal{I}\), then there is a canonical isomorphism

\[ b^{-1}(U)=Proj(\oplus_{d\geq 0}I^d) \]

identifying \(b^{-1}(U)\) with the projective spectrum of Rees algebra \(Bl_I(A)\). Moreover, \(b^{-1}(U)\) have a covering by affine locals \(D_+(a)\) where \(a\in I\), which is identical to spectrum of \(A[\frac{I}{a}]\).

Specifically, according to the definition, blowing up along an empty subscheme \(Z\) we may get \(Proj_X(\mathcal{O}_X[t])\) which is just identical to \(X\). Then using 0804 it’s obvious that a blowup \(b:X'\rightarrow X\) along \(Z\subset X\) restricting to \(X\backslash Z\) is \(id\). From the properties of Rees algebra we know that the exceptional divisor, namely \(b^{-1}Z\) is indeed an effective cartier divisor.

0805

Moreover, blowup commutes with flat base change.

An interesting topic is Raynaud & Gruson’s flatification by strict transformation.

R&G flatification

To be continued

Deformation to the normal cone

Some notations and facts

Following Fulton’s notation, for a graded \(\mathcal{O}_X\)-algebra \(S^\bullet=S^0\oplus S^1\oplus\cdots\) on a scheme \(X\), such that the canonical map from \(\mathcal{O}_X\) to \(S^0\) is an isomorphism, and \(S^\bullet\) is locally generated by sections of degree \(1\), we associate the cone of \(S^\bullet\)

\[ C=Spec(S^\bullet) \]

with canonical projection \(\pi:C\rightarrow X\), and the projective cone

\[ P(C)=Proj(S^\bullet) \]

with canonical projection \(p:P(C)\rightarrow X\).

Let \(z\) be a free variable, \(S^\bullet[z]\) the graded algebra whose nth graded piece is

\[ S^n\oplus S^{n-1}z\oplus\cdots\oplus S^1z^{n-1}\oplus S^0z^n. \]

The corresponding cone is denote \(C\oplus 1\), and the cone \(P(C\oplus 1)\) is called the projective completion of \(C\). Now \(z\) is a section in \(\mathcal{O}(1)\) hence defines a hypersurface which is clearly isomorphic to \(P(C)\), which is then called the hyperplane at infinity. Cutting out this subscheme, the rest part is \(D_+(z)\) and hence the spectrum of algebra \(S^0\oplus S^1/z\oplus S^2/z^2\oplus\cdots\), hence isomorphic to \(C\).

A vector bundlle \(E\rightarrow X\) is the cone associated to the graded sheaf \(Sym(\mathcal{E}^\vee)\), where \(\mathcal{E}\) is the sheaf of sections of \(E\rightarrow X\). The projective bundle of \(\mathcal{E}\) is \(P(E)=Proj(Sym\mathcal{E}^\vee)\). Under this notation, there is a canonical surjetion \(p^*\mathcal{E}^\vee\rightarrow\mathcal{O}(1)\) on \(P(E)\), which identifies \(\mathcal{O}(-1)\) as a subsheaf of \(p^*\mathcal{E}\), which is then called the tautological line bundle of \(P(E)\). Given morphism \(f:T\rightarrow X\), to factor \(f\) through \(p:P(E)\rightarrow X\) is equivalant to specify a line sub-bundle of \(f^*\mathcal{E}\), namely the pullback of the tautological bundle of \(P(E)\).

Let \(X\) be a subscheme of \(Y\), defined by ideal sheaf \(\mathcal{I}\). The normal cone \(C_XY\) is the cone over \(X\) defined by the graded \(\mathcal{O}_X\) algebra \(\oplus_{n\geq 0}\mathcal{I}^n/\mathcal{I}^{n+1}\). This construction commutes with flat base changes. If \(X\rightarrow Y\) is a quasi-regular immersion of codimension \(d\), the graded algebra is then isomorphic to \(Sym(\mathcal{I}/\mathcal{I}^2)\), which is the associated vector bundle of the dual of \(\mathcal{I}/\mathcal{I}^2\). We may denote the cone of \(Sym(\mathcal{I}/\mathcal{I})\) by \(N_XY\). Remember the blowup \(b:Y'\rightarrow Y\) at \(X\) is defined by \(Proj(\oplus_{n\geq 0}\mathcal{I}^n)\), then the pullback of \(X\rightarrow Y\) along the blowup, or namely the exceptional divisor, is then \(\mathcal{O}(1)\), hence a cartesian divisor indeed. If we define by \(E\) the exceptional divisor, then it’s clear that \(\mathcal{O}(E)\) is the dual invertible sheaf of \(\mathcal{O}(1)\), hence isomorphic to \(\mathcal{O}(-1)\).

If \(Y\) is flat over a non-singular curve \(T\), then \(Bl_XY\) is also flat over \(T\) for any subscheme \(X\). Note that being flat over a DVR is equivalant to being torsion free. Assume that \(T\) is a DVR, \(\mathcal{O}_Y\) is therefore flat over \(T\), hence an ideal sheaf \(\mathcal{I}\) is flat over \(T\). Then clearly blowup, which is locally a localization of \(\oplus\mathcal{I}^n\), is flat over \(T\).

Deformation to the normal cone

For a subscheme \(X\) of \(Y\), define \(M_XY\) of simply \(M\) to be the blowup of \(Y\times\mathbb{P}^1\) at \(X\times\infty\). Namely, we has a commutative diagram

The immersion \(i\) is not that straightforward. To construct it, identify \(X\) as the point at infinity of \(X\times\mathbb{P}^1\), which is then embedded into \(Y\times\mathbb{P}^1\). The blowup of \(X\times\mathbb{P}^1\) at infininty is indentical to itself, hence the strict transformation gives the closed embedding \(i\). Since \(pr:Y\times\mathbb{P}^1\rightarrow\mathbb{P}^1\) is flat, the blowup \(M_XY\) is flat over \(\mathbb{P}^1\) with the associated morphism denoted by \(\rho\). Since \(M\rightarrow Y\times\mathbb{P}^1\) is an isomorphism if we cut out \(Y\times\infty\), we see that

• \(i\) is the trivial embedding away from \(\infty\), namely \(X\times\mathbb{A}^1\rightarrow Y\times\mathbb{A}^1\).

In the theory of \(A^1\)-homotopy, we hope to do such a deformation on section \(0\) and \(1\) over an affine line (dispite that, it’s very stupid to do calculation over a projective base!). So here we cut off the \(0\) point of \(\mathbb{P}^1\) and identify it as \(\mathbb{A}^1\) with \(\infty\) as \(0\). We denote by \(M_XY'\) be \(M_XY\) localized over \(\mathbb{A}^1\).

If we denote by \(\mathcal{I}\subset\mathcal{O}_Y\) the ideal cutting out \(X\) inside \(Y\), then the ideal cutting out \(X\times 0\) in \(Y\times\mathbb{A}^1\) is just \((\mathcal{I}[T]+(T))\subset\mathcal{O}_Y[T]\), so the morphism \(M_XY\rightarrow Y\times\mathbb{A}^1\) is \(Proj_{Y\times\mathbb{A}^1}(\oplus_{n\geq 0}(\mathcal{I}[T]+(T))^n)\). Then what’s the exceptional divisor? We may illustrate this using some small tricks…

Going in (upper) vertical direction stands for \(\mathcal{I}\) filtration and (right) horizental direction stands for \((T)\) filtration. Now this table depicts the ideal \(\mathcal{J}:=\mathcal{I}[T]+(T)\). Then \(\mathcal{J}^n\) is just the diagram with a size \(n\) triangle at lower left corner, and the quotient \(\mathcal{J}^n/\mathcal{J}^{n+1}\) is

(for the case \(n=2\)). Namely this is \(\oplus_{0\leq m\leq n}(\mathcal{I}^m/\mathcal{I}^{m+1})(T^{n-m})\), so the exceptional divisor of \(M_XY'\) localized over \(\mathbb{A}^1\) is just \(P(C\oplus 1)\) where \(C\) is the normal cone \(C_XY\).

Now the only trouble is the fibre of \(M_XY'\) over \(0\) (or \(\infty\) in our previous language). Namely this is the projective cone of \(\oplus\mathcal{J}^n/T\mathcal{J}^n\) over \(Y\), so using the depiction above we know that the slices of this graded algebra looks like

(\(n=2\) e.g.). Namely the slice at degree \(n\) is \((\mathcal{I}^n\oplus(\mathcal{I}^{n-1}/\mathcal{I}^n)T\oplus\cdots\oplus(\mathcal{O}_Y/\mathcal{I})T^n)\). This projective cone has two obvious subschemes: the one associated to the left column, namely the blowup of \(Y\) at \(X\), and the tilt slice, namely \(P(C\oplus 1)\) as what we has illustrated above. These two subschemes meet at the projective cone \(P(C)\) which relates to the first non-empty block on the first column. Furthermore, these two closed subschemes are both cartier divisors if we view the fibre as a closed subscheme of \(M_XY'\). This is obvious for the exceptional divisor, and it’s straightforward that the embedded blowup is \((T)\) where \(T\) viewed as a section of \(\mathcal{O}(1)\).

The next step is to drop the blowup part \(Bl_XY\) inside \(M_XY'\).