Blow up of a Projective scheme along a projective-subscheme using Macaulay2

Question

Given a closed-subscheme (corresponding to the homogeneous ideal $I$ inside $K[x_0,...,x_n]$) of a projective scheme $\mathbb{P}^n$, how can one find out the blow up using Macaulay2? Proj(reesAlgebra(I)) gives the blow-up of $I$ along with an affine scheme. Is there any function that we can use? Or we need to do something else?

Answer 1

Let $S=k[x_0,\ldots,x_n]$ the polynomial ring and $I=(f_1,\ldots,f_s)$ be the ideal to be blown up. Then the blowup $X$ of $\mathbb{P}^n$ at $I$ is

$$ \mathrm{Proj}(S \oplus I \oplus I^2 \oplus \cdots) = \mathrm{Proj}(T) $$

with $T = S[t I] \subseteq S[t]$ a subring of $S[t]$ ($t$ a new indeterminate). (Proj here in the sense of Hartshorne, Algebraic Geometry, p.160, of an S-Algebra)

Now we have a surjection $\phi:S[T_1,\ldots,T_s] \to S[t I]$ with $\phi:T_i \mapsto t f_i$.

Setting up the map of polynomial rings $\phi:S[T_1,\ldots,T_s] \to S[t]$ is easy in Macaulay2 and it is equally easy to compute $\ker \phi$ (it is an inbuilt operation). So $T$ can be found explicitly as $T = S[T_1,\ldots,T_s]/(\ker\phi)$. To make this the proj of a ring $R$ over $k$ consider the map $$ \psi:R = k[u_{ij}] \to S[T_1,\ldots,T_s], \quad u_{ij} \to x_i T_j $$ where $0 \leqslant i \leqslant n$ and $1 \leqslant j \leqslant s$.

Use this map to form a map $\psi':R \to T$ and compute $J = \ker \psi'$. Then $\mathrm{proj}(R/J)$ is isomorphic to the blowup $X$ from the beginning and is explicitly represented as a closed subscheme of $P^N$ with $N=(n+1) s -1$. (This is an application of the Segre-embedding).

It is possible to extend this discussion to the case of $S=k[x_0,\ldots,x_n]/\mathfrak{a}$ where $\mathfrak{a}$ is an ideal of $k[x_0,\ldots,x_n]$ and to do essentially the same computation with Macaulay2 getting the blow-up of $V(\mathfrak{a})$ at an ideal $\mathfrak{a} + I$.