Definition: For a finite subset $X \subset \mathbb P^r$, the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in $X$, usually called the degree of $X$.

Let $I_d(X)=(x_0^2,x_0x_1,x_1^2,x_0x_2^d-x_1x_3^d) \subset k[x_0,x_1,x_2,x_3]$. Compute the regularity and degree of $X$.

Note that $(x_0^2,x_1^2) \subset I_d(X) \subset (x_0,x_1)$, hence $\sqrt{I_d(X)}=(x_0,x_1)$. Therefore the local cohomology modules supported in $I_d(X)$ and $(x_0,x_1)$ are isomorphic. Hence if I have computed correctly I think $\mathrm{reg}(I_d(X))=1$. Is this correct? And also how do we compute $\deg(X)$ in practice?

  • $\begingroup$ I personally think that the degree of $X$ and the degree of $X^{\operatorname{red}}$ are not the same thing. The definition you give for degree is only the correct one if $X$ is a reduced finite subscheme. (In general, it is still given by the Hilbert polynomial, but this is not equal to the number of points.) For example, if $X$ is given by $(x_0^2) \subseteq k[x_0,x_1]$, then this should have degree $2$, even though there is only one point (namely $[0:1]$). $\endgroup$ – Remy May 6 '16 at 4:11

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