bio | website | math.univ-toulouse.fr/… |
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location | Toulouse | |
age | 43 | |
visits | member for | 3 years, 3 months |
seen | yesterday | |
stats | profile views | 2,135 |
Chercheur au CNRS, working in Toulouse.
I don't usually comment or answer questions from anonymous users.
Nov 19 |
accepted | On a proposition in Hartshorne's paper “Ample vector bundles on curves” |
Nov 16 |
awarded | Nice Question |
Nov 16 |
comment |
On a proposition in Hartshorne's paper “Ample vector bundles on curves”
Thank you very much for taking the time to write this answer. If I understand your answer right, your construction is in the same line as what ulrich described in his comment. I would still like to see a proof or a counterexample to Prop. 4.1, though but if nobody provides one in the next few days I will accept your answer. |
Nov 15 |
comment |
On a proposition in Hartshorne's paper “Ample vector bundles on curves”
@ulrich. Thank you very much for this interesting comment. That settles it then, as far as (X) is concerned. Here is a possible simplification of your argument: as you write, $C\to f(C)$ must be generically separable because $H^0(A,\Omega_A)\to H^0(C,\Omega_C)$ does not vanish. Now since $C\to f_*(C)$ is also generically inseparable, it must be birational. Note that $f_*(C)$ might be singular, so its geometric genus might not be defined (or is there a way to show that it must be non-singular ?). |
Nov 14 |
comment |
On a proposition in Hartshorne's paper “Ample vector bundles on curves”
Thank you for this reference. I had a look at Fulton's paper but I cannot find an argument in there that shows that (X) (in my post) or Prop. 4.1 is wrong in a specific situation. The counterexample of Serre that Fulton mentions is apparently a construction that appears in the very paper by Hartshorne that I am concerned with (in par. 3). It is an example of a curve and a vector bundle on it, which is not ample and yet has only positive quotients. How does all this connect to (X) or Prop. 4.1 ? |
Nov 14 |
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On a proposition in Hartshorne's paper “Ample vector bundles on curves”
@Donu Arapura: $\phi$ is birational onto its image (I edited the question 30 minutes ago - you might have read the post just before that - sorry). |
Nov 14 |
asked | On a proposition in Hartshorne's paper “Ample vector bundles on curves” |
Aug 25 |
comment |
Rational points techniques on curves not using their Jacobian
The method of Kim is very interesting but he can only prove Siegel's theorem for the projective line, as far I can see. In that case, (much) more elementary proofs can be made quite effective (see Silverman's book on elliptic curves) but even this does not give a method to find all the rational points on a curve. |
Aug 24 |
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Rational points techniques on curves not using their Jacobian
The very conjectural "effective Mordell conjecture" (see Astérisque 183) would provide such a method. The analog of this conjecture is proven over function fields (proofs by Arakelov and Szpiro) and such a method is thus available over function fields. Over number fields, I don't know any method that does not use the Jacobian. |
Aug 24 |
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Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$
What is $X_{\rm can}$ exactly ? Also, is it clear that $\Omega_{X/X_{\rm can}}$ is locally free ? If so, why is $f_*(\Omega_{X/X_{\rm can}})$ locally free ? If it is not, then what do you mean by hermitian metric on that object ? (sorry if all this is standard). |
Aug 22 |
awarded | Yearling |
Jul 2 |
awarded | Curious |
Jun 20 |
comment |
splitting of the Hodge filtration
I think the OP means: "When is the canonical Hodge splitting already defined over $k$ ?". This happens rarely and the failure to split over $k$ is basically encoded in the Mumford-Tate group, about which there are many deep conjectures. For instance if $X$ is a CM abelian variety and $k$ contains the Galois closure of the CM field, then the sequence splits (and the Mumford-Tate group is a torus). |
May 14 |
comment |
Ampleness of Hodge bundles over complex curves
@Jason Starr. Thank you for your remark. I would expect something like that but I cannot find any coherent bibliographical reference for this kind of thing. |
May 14 |
asked | Ampleness of Hodge bundles over complex curves |
Apr 1 |
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Are there perverse sheaves on abelian varieties with small Euler characteristic?
Isn't $d=g$ in your last statement ? |
Apr 1 |
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Is it possible to prove Mordell's conjecture geometrically?
This is a good question. It has always struck me that the only known proofs of the Mordell conjecture over number fields, as well as of the Manin-Mumford conjecture require arithmetic models. The only thing I know in the "purely complex" (or "geometrical") direction is Zannier-Pila's approach to Manin-Mumford (but not Mordell...), which still requires arithmetics, but less so than Raynaud's (or Ullmo-Szpiro-Zhang, or Pink-R.'s). |
Mar 23 |
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Chow ring of a $\mu_2$-gerbe
I have only recently come across your question. Do you actually mean a $\mu_2$-torsor ? |
Mar 23 |
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Inequality regarding sum of gaussian on lattices
It might be interesting to consider the special case where the lattice comes from the fractional ideal of a number field. Then you could use the functional equation of the theta function. See arxiv.org/pdf/math/9802121v3.pdf or math.univ-toulouse.fr/~rossler/mypage/pdf-files/… |
Mar 21 |
comment |
A naive algebraic geometry question
See Hartshorne, Ex. III.4.2, p. 222 (Chevalley's theorem). |