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I was trying curious to follow up the discussion from: Normal quadric surfaces in $\mathbb{P}^3$

which is exercise I.3.17(b), Hartshorne, for the case of characteristic $2$.

So consider the ring extension $k[x,y] \hookrightarrow R = {\frac {k[x,y,z]}{ \langle x^2 - yz \rangle }}$ and let $Q(R)$ denote the quotient field of $R$. (Take $k = {\overline{ {\mathbb{F}}_2 }}$.)

${\textbf{Question.}}$ Is $R$ integrally closed?

If I assume that $f_1/{f_2} + g_1/{g_2} \cdot z$ is integral over $R$ with $f_i, g_i \in k[x,y]$, then we get the condition ${\mathrm{g.c.d.}}(f_2, g_2) \neq 1$. However, it looks tricky to produce a counter example.

Siddhartha
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  • It satisfies the Serre conditions $R_1$ and $S_2$ and hence normal. – Mohan Jul 06 '20 at 01:50
  • Can you give a reference? – Siddhartha Jul 06 '20 at 01:53
  • @Siddhartha googling "Serre's criteria for normality" provides many resources - Wikipedia, StacksProject, etc. You can also verify that $R$ is integrally closed directly from the definition in this case, but it can be a little onerous. – KReiser Jul 06 '20 at 02:43
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    For which statement? That Serre conditions imply normality or your ring satisfies Serre conditions ? – Mohan Jul 06 '20 at 02:44

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