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I am trying to solve the following, but I am stuck. Let $P=\mathbb{P}_k^d $ be a projective space over a field k. Let $X$ be a complete subvariety of $P$ of dimension $r$. Say that X is a complete intersection if it is defined by $d-r$ homogenois polynomials $F_1,..., F_{d-r}$ . Let $Z=V_+(F_1,..., F_{d-r-1})$ and consider the closed subscheme of $Z$ corresponding to $V_+(F_{d-r})$ in $Z$. How could I show that the ideal sheaf of $Z$ cutting out $X$ is isomorphic to $\mathcal{O}_Z(-\deg F_{d-r})$?

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Consider the case $r=1.$ Here we have the short exact sequence $0\to(F_1)\to A\to A/(F_1)\to 0,$ where $A = k[x_0,\ldots, x_d],$ which sheafifies to $0\to\mathcal I_X\to\mathcal O_{\mathbb P^n}\to\mathcal O_X\to 0.$ It is well known that $\mathcal I_X\cong\mathcal O_{\mathbb P^n}(-X),$ considering $X$ as a Cartier divisor on projective space (e.g. Hartshorne Prop. II.6.18), and since $\operatorname{Pic}(\mathbb P^n)\cong \mathbb Z$ by the degree map, we know that $\mathcal O_{\mathbb P^n}(-X)=\mathcal O_{\mathbb P^n}(-\deg F_1).$

A similar analysis works in general.

We have a short exact sequence $0\to (F_{d-r})\to A\to A/(F_{d-r})\to 0,$ where $A$ is the (projective) coordinate ring $A = k[x_0,\ldots,x_d]/(F_1,\ldots, F_{d-r-1}).$ Sheafifying gives $0\to\mathcal I_X\to\mathcal O_Z\to\mathcal O_X\to 0.$ Since $X$ has dimension $r$ in $\mathbb P^n,$ it has codimension $1$ in $Z.$ Moreover, it is clearly locally principal (defined by $F_{r-d}$), and thus is a Cartier divisor on $Z.$ So again, we find that $\mathcal I_X\cong\mathcal O_Z(-X).$ In this case, notice that $\mathcal O_Z(-X)\cong\mathcal O_Z\otimes\mathcal O_{\mathbb P^n}(-X) = \mathcal O_Z\otimes\mathcal O_{\mathbb P^n}(-\deg F_{d-r})\cong\mathcal O_Z(-\deg F_{d-r}).$

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