Timeline for Regular hypersurface containing a point of a variety $X$ over perfect field $k$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 24, 2023 at 16:23 | comment | added | Kapil | Let $C$ be an irreducible curve which is singular at $p$. We look at $X=C\times C$ and $x=(p,p)$ in $X$. Then every curve in $X$ that passes through $x$ will be singular. | |
| Mar 24, 2023 at 14:59 | comment | added | JackYo | In your last sentence sentence you mentioned that one cannot put a regularity condition for the ambient scheme only at the point and ask for global regularity of the divisor. I'm confused, in original question I even not assumed that the ambient space $X$ should be smooth at $x$. So you suggest, that in such case is not reasonable to ask if $x$ is contained in a smooth hyperplane? Is it wrong? So it can happen that for reduced $x \in X$ with base field $k$ perfect, the point $x$ is NOT contained in a smooth hypersurface? Do you know a counterexample? | |
| Mar 24, 2023 at 14:53 | comment | added | JackYo | The global question is about local equations of effective divisors (since if we assume $X$ to be smooth, therefore normal, so Weil divs= Cartier divs), and therefore about smoothness of general member of a linear system. For ample linear systems that's essentially Bertini | |
| Mar 24, 2023 at 14:31 | comment | added | JackYo | @Kapil: the first one looks doable, if $R= \mathcal{O}_{X,x}$ is the stalk on $x$, then one can take as divisor $V(f)$ for $f \in \mathfrak{m}_x-\mathfrak{m}_x^2$. If we assume $R$ be regualr, then $R/f$ is regular in $x$ and the regularity of $V(f)$ extends to an open nbhd of $x$. | |
| Mar 24, 2023 at 14:24 | history | edited | JackYo | CC BY-SA 4.0 |
added 102 characters in body
|
| Mar 24, 2023 at 14:07 | comment | added | Kapil | There are two slightly different questions that you can ask: (Local question) Given a regular point on a scheme is there a divisor that contains this point which is smooth at that point. (Global question) Given a smooth variety over a perfect field and a point of this variety, is there a smooth hypersurface that contains this point. One cannot, as you appear to be doing, put a regularity condition for the ambient scheme only at the point and ask for global regularity of the divisor. | |
| Mar 24, 2023 at 14:04 | history | edited | JackYo | CC BY-SA 4.0 |
added 119 characters in body
|
| Mar 24, 2023 at 13:43 | history | asked | JackYo | CC BY-SA 4.0 |