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

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?

## 1 Answer

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 $$\mathfrak{a} + I$$.

• In that case what would be the from proj(R/J) -> P^n? – Anko Nov 7 '18 at 1:55