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715
bio website math.univ-toulouse.fr/…
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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
comment 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
comment 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
comment 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
comment Are there perverse sheaves on abelian varieties with small Euler characteristic?
Isn't $d=g$ in your last statement ?
Apr
1
comment 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
comment 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
comment 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).