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Damian Rössler

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Name Damian Rössler
Member for 1 year
Seen 14 hours ago
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Location Toulouse
Age 42
Chercheur au CNRS, working in Toulouse. I don't usually comment or answer questions from anonymous users.
Jun
8
 comment Belyi’s theorem for function fields
(answering your comment below). I think that it is unlikely that this will work if you want the morphism to be tamely ramified. Galois coverings of the affine line in char. p, which are tamely ramified along $\infty$, are trivial.
Jun
7
 accepted Belyi’s theorem for function fields
Jun
7
 comment What analysis should I know for studying Arakelov Theory?
Global Arakelov theory requires a lot of index theory. Some of that material is covered in the book Heat kernels and Dirac operators, by N. Berline, E. Getzler and M. Vergne, Grundlehren Math. Wiss., vol. 298, Springer-Verlag, New York, 1992. The difficult part of local index theory, which is needed in the proof of the arithmetic Riemann-Roch theorem is only described in the articles of Bismmut and his coworkers (the most self-contained one is his article with Lebeau).
Jun
7
 comment What analysis should I know for studying Arakelov Theory?
You could have a look at C. Soulé book "Lectures on Arakelov Geometry" (Cambridge Univ. Press).
Jun
7
 answered Belyi’s theorem for function fields
Jun
4
 comment Duality and Hirzebruch-Riemann-Roch
There is a link between duality for coherent sheaves and Grothendieck-Riemann-Roch. Consider the Lefschetz-Verdier formula for coherent sheaves (see SGA 5, Exp. III, Appendix) and (with the terminology of the appendix) let $X_1=X_2=X$, $Y_1=Y_2=Y$ and $C'=C''=$diagonal, $D'=D''$=diagonal. Then the formula gives (although this is difficult to show...) Grothendieck-Riemann-Roch with values in Hodge cohomology, ie $\oplus_{p}H^p(Y,\Omega^p)$. The Lefschetz-Verdier formula depends only on duality.
May
19
 comment Connectedness of hyperplane sections (reference request)
( Hartshorne, Algebraic Geometry, Springer, GTM 52, Cor. 7.9, p. 244 )
May
16
 comment Identity on topological space but not on scheme
Notice that an automorphism of a field will also do the trick.
May
16
 comment Identity on topological space but not on scheme
($k$ is a field in my last comment).
May
16
 comment Identity on topological space but not on scheme
There is the composition $k[\epsilon]/\epsilon^2\to k\to k[\epsilon]/\epsilon^2$, where the first arrow is the morphism of $k$-algebras sending $\epsilon$ to $0$ et the second one is the morphism making $k[\epsilon]/\epsilon^2$ into a $k$-algebra.
May
10
 comment What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?
There is the first chapter of the book "Sheaves on manifolds" by Kashiwara-Shapira.
May
9
 comment Shafarevich’s theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field
I think the following problem is lurking in the background here: the action of the automorphism group of $K$ on $j$. In order to get finiteness statements, this group should be seen to be finite (eg when $K$ is the function field of a curve over a finite field).
May
6
 comment Lefschetz hyperplane section theorem for connections
@ulrich: I see - you are right of course.
May
6
 comment Lefschetz hyperplane section theorem for connections
... what I wrote above does not provide a counterexample but it might suggest how to construct one (or provide a proof !).
May
6
 comment Lefschetz hyperplane section theorem for connections
By Cor. II.6.11 in Deligne's "Equations différentielles..." (LNM 163), the cohomology groups you want to compute are the cohomology groups of the corresponding local systems (for the ordinary topology). Now notice that the topological proof of Lefschetz's theorem is based on the result that a non-singular $k$-dim. affine variety has the homotopy type of $k$-dim. CW complex. So you are led to the question: does the cohomology of a local systems on a $k$-dim. CW complex vanish above $2k$ ? This is unlikely to be true in general (classifying spaces probably provide counterexamples).
May
2
 comment reference request for the finiteness of cuspidal subgroup of $X_0(N)$?
See also R. Elkik, "Le théorème de Manin-Drinfeld." Astérisque, pages 59-67, 1990. Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988). The proof given there follows an idea of Deligne and uses mixed Hodge structures.
Apr
29
 comment Zariski’s main theorem in the form of Grothendieck, universal properties
I don't think that (3) holds. This would imply that every ${\bf Z}/n$-torsor on $W^\circ$ can be extended to a ${\bf Z}/n$-torsor over $W$. There is an obstruction for this to be possible, which lies in a $H^2$ group.
Apr
28
 comment Irreducibility of superelliptic curves
Dear Felipe, I was about to ask you about (b). If I am not mistaken, a) and b) have positive answers even only for $d>0$ (one can apply Eisenstein's criterion in $(k[x])[Y]$).
Apr
11
 comment (Open) weak Lefschetz with coefficients
If you want log singularities, you could perhaps proceed like this: first look up the results of Esnault-Viehweg on generalizations with log. sing. of the Kodaira vanishing theorem and of the degeneration of the Hodge-de-Rham spectral sequence (maybe in their book ?) and then replicate the proof of the weak Lefschetz theorem given in the book of Griffiths and Harris (p. 156, chap. I, §2).
Apr
11
 comment Closed subschemes and the analytification functor
Have you had a look at the beginning of Serre's GAGA paper ? There is a very down to earth description of the analytification functor there.
Apr
11
 comment scheme of generalizations
The set $U$ is not constructible in general (so not in particular not a subscheme). From the point of view of formal geometry, it is a closed subset but it is certainly not a Zariski closed subset in general.
Apr
9
 comment Elliptic curve over a scheme is a group scheme?
Sorry I just noticed that pranavk made a similar comment... I am leaving my comment because it contains the corresponding bibliographical reference.
Apr
9
 comment de Rham complex of closed immersion between smooth schemes
The mistake is the fact that the differentials of the de Rham complex are not $Q-$ (resp. $P-$) linear and thus you cannot apply the fact that flat morphisms preserve homology.
Apr
9
 comment Elliptic curve over a scheme is a group scheme?
There is a result of Grothendieck, proven in Mumford's GIT book (Th. 6.14, p. 124 in the third edition), which says that a smooth proper morphism, which has one fibre, which is an elliptic curve, can be endowed with the structure of an abelian scheme. So if one fibre of your family has a rational point, the two notions will coincide... (in particular if there is a section).
Apr
9
 comment relative resolution of singularity
For families of curves, you might find the following article interesting: B. Teissier, "Résolution simultanée I: famille de courbes" (available on Numdam). There is also "Résolution simultanée II" in "Séminaire sur les singularités des surfaces" (1976) but I don't know whether it is available in digital form.
Apr
9
 comment smooth algebras and triviality of de Rham complex
This is not so. Let $R={\bf Q}$. In the situation you describe, the cohomology of the de Rham complex $\otimes{\bf C}$ is then actually the singular cohomology of the space ${\rm Spec\ A}({\bf C})$. This is a theorem of Atiyah-Hodge-Grothendieck. See "On the de Rham cohomology..." by A. Grothendieck, Th. 1. In general, you may find an $A$ so that the singular cohomology of the space ${\rm Spec A}({\bf C})$ does not vanish. In the case of ${\bf C}[T_1,\dots, T_n]$, it holds because affine space is contractible.
Apr
8
 comment surjective submersion and fibrations
If the submersion is proper then this is Ehresmann's theorem. See en.wikipedia.org/wiki/Ehresmann's_theorem
Apr
5
 comment Algebraic $p$-adic integers mod $p$
Since $(p)$ is the generator of the maximal ideal of ${\bf Z}_p^{\rm nr}$, you have ${\bf Z}_p^{\rm nr}/(p)=\bar{\bf F}_p$.
Apr
5
 comment sections of vector bundles transversal to a divisor
A section of $E$ cannot meet a point in $D$ transversally unless it avoids it. Is this what you have in mind ?
Apr
2
 comment How to define intersection of coherent sheaf and 1-cycle?
I don't think you can. This is reflected in the fact that Chow homology in the sense of Baum-Fulton-MacPherson does not have a ring structure. Notice also that the Grothendieck group of coherent sheaves does not have a ring structure in general.
Mar
23
 comment Six operations for (quasi)-coherent sheaves
There is no lower shriek functor in the category of quasi-coherent sheaves, unless the morphism is proper. See the appendix by Deligne to Hartshorne's "Residues and Duality" (you have to expand the category to get a good lower shriek).
Mar
14
 comment On the square theorem for commutative algebraic groups
Not all commutative algebraic groups satisfy the theorem of the cube. See the first chapter of Moret-Bailly's book "Pinceaux de variétés abéliennes".
Jan
27
 awarded ● Nice Answer
Jan
15
 answered the dual abelian scheme