Affine cone example

Question

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\displaystyle {\mathcal {I}}=(f)(g_{1},g_{2},g_{3})}$.

How to calculate that

$${\displaystyle \bigoplus _{n\geq 0}{\frac {{\mathcal {I}}^{n}}{{\mathcal {I}}^{n+1}}}\cong {\frac {{\mathcal {O}}_{X}[a,b,c]}{(g_{2}a-g_{1}b,g_{3}a-g_{1}c,g_{3}b-g_{2}c)}}}$$

is an isomorphism of ${\mathcal {O}}_{{{\mathbb {P}}^{n}}}$-modules?

Source: example for graded algebra of affine cone.

Answer 1

The algebra that you are describing gives the fibre over $X$ in the blow-up of $\mathbb{P}^n$ along $X$. (The blow-up is given by $\mathrm{Proj}(\bigoplus_{n=0}^\infty I^n$.) Therefore, in order to prove the isomorphism in question, one can have the following strategy; details are found, for example, in the book Integral closure of ideals, rings and modules by Swanson and Huneke, in the chapter on Rees algebras.

Write $R = \mathbb{C}[x_0, \ldots, x_n]$ and $I$ for the ideal in $R$ (abusing notation). The algebra $\bigoplus_{n=0}^\infty I^n$ can be written as a quotient of $R[a,b,c]$ with $a \mapsto fg_1, b \mapsto fg_2, c \mapsto fg_3$. There are the immediate relations: $g_2a - g_1b, g_3a-g_1c, g_3b-g_2c$; these are exactly the relations coming from the syzygies of $fg_1, fg_2, fg_3$, since, while computing the syzygies of these three polynomials, we may as well take $f=1$. Then the kernel of the above quotient map is given by $(g_2a - g_1b, g_3a-g_1c, g_3b-g_2c) : (abc)^\infty$, (the saturation of the the ideal $(g_2a - g_1b, g_3a-g_1c, g_3b-g_2c)$ by the element $abc$). But this situation is the same as the one in which $f=1$, i.e., we are computing the Rees algebra of the regular sequence $g_1, g_2, g_3$. (See, e.g., Hartshorne, Algberaic Geometry, Chapter II, Exercise 7.11(b) for the geometric version.). It is known that for a regular sequence $g_1, g_2, g_3$, the defining ideal of the Rees algebra is $(g_2a - g_1b, g_3a-g_1c, g_3b-g_2c)$.