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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.

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