Timeline for Normal crossings on a surface and ordinary double points
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10 events
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| Feb 22, 2011 at 3:33 | comment | added | EricB | Dear Sándor: I think I am being dense here, but I still don't see it. Suppose $R=Z_p$, and at a point $x$ we have $X_0=div(fg)$, where $f,g$ is a system of parameters in $O_{X,x}$. This ring $O_{X,x}$ is a regular 2-dimensional local ring, so its completion is in general an Eisenstein extension of some $R'[[u,v]]$. But I cannot see how to say that it is $R[[f,g]]$, so I don't see how to say that the completion of the closed fiber ring is $F[[f,g]]/(fg)$, which is the definition I'm trying to use for ordinary double point. It must be something obvious, I'll think about it. Thanks! | |
| Feb 22, 2011 at 3:16 | comment | added | Karl Schwede | Of course, you can have something irreducible and not normal crossings such that when you base change it appropriately, you get simple normal crossings. Consider $\mathbb{R}[x, ix]$ base changed over $\mathbb{R}$ with $\mathbb{C}$. The output is $\mathbb{C}[a,b]/(a^2 + b^2)$ which is obviously simple normal crossings. | |
| Feb 22, 2011 at 3:15 | comment | added | Karl Schwede | EricB, "but isn't there the same problem in either case to translate to an ordinary double point?" I don't think there is any problem translating my alternate definition as long as the residue fields at closed are algebraically closed (you shouldn't really need this) and the ring is excellent. As Sándor said, that case is basically covered in I, Exercise 5.14 in Hartshorne. | |
| Feb 22, 2011 at 3:10 | comment | added | Sándor Kovács | By the way, it turns out that having normal crossings is actually also an intrinsic property, but it is not obvious from the definition. Just like being a complete intersection is. :) | |
| Feb 22, 2011 at 3:05 | comment | added | Sándor Kovács | The last statement in my previous comment follows from the fact that if the completion of a local ring is isomorphic to the completion of the local ring of an ordinary double point of a curve, then the original local ring was the local ring of an ordinary double point. This is an exercise somewhere near the end of the first chapter of Hartshorne. | |
| Feb 22, 2011 at 3:03 | comment | added | Sándor Kovács | Eric, I almost made the same remark as Karl regarding normal crossings versus simple normal crossings. The difference is whether you allow self-intersections of the irreducible components or not. As far as the singularities are concerned, they are ordinary double points in each case. In other words, normal crossings only asks for a certain behaviour in an analytic or formal neighbourhood while simple normal crossings ask for that in a Zarsiki neighbourhood. As far as ordinary double points are concerned, there is no difference. | |
| Feb 22, 2011 at 2:27 | comment | added | EricB | Thanks for the comment Karl. I'm using a definition in Grothendieck-Murre (Tamely ramified covers...), which is essentially compatible with one in the AG book by Q. Liu. It does not speak of smooth, I believe, only regular. I guess the point I'm interested in is whether Lipman's embedded resolution theorem can produce a relative curve such that $(X_0)_red$ has only ordinary double points for singularities, hang the normal crossings! Especially About your other comment, I think I've seen that definition, but isn't there the same problem in either case to translate to an ordinary double point? | |
| Feb 22, 2011 at 2:23 | comment | added | Karl Schwede | I think it depends on how you define normal crossings though. Another definition I've seen is that two curves meeting at a point $x \in X$ having normal crossings means that the two curves are smooth at $x$ and cross at $x$ transversally, and that point $x$ is a regular point (or maybe smooth point depending on the context) of $X$. Of course, if $X$ is not smooth at a point $x$, you could still have two smooth curves crossing transversally at that point (for example, the two rulings of a quadric cone), but it wouldn't be called normal crossings. | |
| Feb 22, 2011 at 2:20 | comment | added | Karl Schwede | EricB, in terms of the definition for normal crossings, I often see simple normal crossings for the definition you gave, if it holds at all points, to avoid this ambiguity. In particular, the definition you gave rules out the nodal curve (in a smooth ambient space) having normal crossings at the node point. Although locally analytically, it certainly does have normal crossings (via your definition). | |
| Feb 22, 2011 at 1:57 | history | answered | EricB | CC BY-SA 2.5 |