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$\underline {Background}$:We know that,for a projective variety $X \subset\mathbb{P}^{n}=(\mathbb{K}^{n+1}-{0})/\sim$

we define , degree($X$)=$(r!)$.(leading coefficient of the hilbert polynomial of $X$)

$\underline {Question (1)}$:What is the definition of degree of a closed subscheme $X$ of $Proj(K[x_0,....,x_n])$?

Can we define the same thing for closed subscheme?

$\underline {Question (2)}$:what can be said about degree of $0$-dimensional subcsheme?

Since there are only a finitely many points in a $0$ dimensional subscheme ,can we say that in this case degree is same as cardinality?

Finally is there any reference where they talk explicitly about the definition of degree of a closed subscheme in $Proj(K[x_0,....,x_n])$(maybe with some example)

Any help from anyone is welcome.

HARRY
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1 Answers1

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Answer (1): yes, we can!

Answer (2): Let $I\subset\mathbb{K}[x_0,\dots,x_n]$ be an homogenous ideal, such that $V_+(I)$ is a $0$-dimensional (closed) subscheme of $\mathbb{P}^n_{\mathbb{K}}$ which support is a set of $r$ distinct points; where the underground field is algebraically closed. One proves easily that $\deg V_+(I)\geq\deg V_+(\sqrt{I})=r$, because $I\subseteq\sqrt{I}$, but in general the equality does not hold.

Example: Let $n=1, I=(x_1)^2=(x_1^2), \sqrt{I}=(x_1)$ and the support of $V_+(I)$ is a single (closed) point of $\mathbb{P}^1_{\mathbb{K}}$; using the proof given here: \begin{equation*} \deg V_+(I)=\dim_{\mathbb{K}}\mathbb{K}[x_1]_{\displaystyle/(x_1^2)}=2>1=\dim_{\mathbb{K}}\mathbb{K}[x_1]_{\displaystyle/(x_1)}=\deg V_+(\sqrt{I}) \end{equation*}

As reference, I suggest the Gathmann's lecture notes (classical Algebraic Geometry and Scheme Theory).

Armando j18eos
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