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May
29
comment Automorphisms of manifold vs automorphisms of Hodge structure
@quinque Thank you for your comment. I think the period domain is 15-dimensional (as the intermediate Jacobian is 5-dimensional); see p. 11 of math.ens.fr/~debarre/Periods-MSRInotes2.pdf
May
29
comment Automorphisms of manifold vs automorphisms of Hodge structure
@quinque I'm not sure if there are automorphisms (besides -1) of the intermediate Jacobian of a cubic threefold which don't come from the cubic threefold itself. Do you think that for a cubic threefold $X$, we have $Aut(X) x \{\pm id\} = Aut(J_X)$?
May
29
revised Automorphisms of manifold vs automorphisms of Hodge structure
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May
28
revised Automorphisms of manifold vs automorphisms of Hodge structure
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May
28
comment Automorphisms of manifold vs automorphisms of Hodge structure
@quinque you're right. Let me edit my answer.
May
27
awarded  Organizer
May
27
revised Automorphisms of manifold vs automorphisms of Hodge structure
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May
27
suggested approved edit on Automorphisms of manifold vs automorphisms of Hodge structure
May
27
answered Automorphisms of manifold vs automorphisms of Hodge structure
May
18
comment Some questions about reduction of elliptic curves
Seems like you're asking about the reduction of $X_0(n)$. Have a look at Katz-Mazur's book Arithmetic moduli of elliptic curves and Brian Conrad's article in J. inst. Jussieu with a similar title.
May
15
comment Some questions about reduction of elliptic curves
Any elliptic curve $E$ over a dense open $U \subset \mathbb A^1_{\mathbb Q}$ extends to an elliptic curve over a dense open $\mathcal U \subset \mathbb A^1_{\mathbb Z}$. That should help you do what you want for large $p$. You won't find a curve over $S$ whose fiber at $s$ is isomorphic to $\tilde{E}_s$ simply because $\mathbb Q$ and $\mathbb F_p$ are not isomorphic...
May
15
comment Some questions about reduction of elliptic curves
Consider an elliptic curve $E \to \mathbb A^1_{\mathbb Q} - \{0,1\}$. Let $s$ be a rational point of the base. Then $E_s$ is an elliptic curve over $\mathbb Q$. What does it mean to "reduce $E_s$" in this case? And do you mean by "glueing everything to a curve over $S$"? Isn't $E\to S$ already the object you need?
May
15
comment Some questions about reduction of elliptic curves
I still don't understand 4. Is $E\to S$ an elliptic curve or is only its generic fibre an elliptic curve? Also, what do you mean by finding the Neron model of $E_s \to Spec(k(s))$? To speak of a model for $X/k$ over a scheme $S$, we usually ask $k$ to be function field of this scheme $S$. Are you asking about whether an elliptic curve $E\to S$ is uniquely determined by its generic fibre? In that case, the answer is positive if $S$ is integral noetherian and regular.
May
15
comment Some questions about reduction of elliptic curves
Have a look at Proposition 1.2.4 in the book "Neron models" by Bosch et al. This proposition implies that, if $N$ is the Neron model of the elliptic curve $E$ over $\mathbb Q$, then $N_{\mathbb Z_{(p)}}$ is the Neron model of $E$ over $\mathbb Z_{(p)}$. I agree that the unicity is not all you need here. To prove that Proposition you need to use limit arguments and reduce to the statement that "the Neron model commutes with etale base-change".
May
12
comment Some questions about reduction of elliptic curves
Yes to 1). The Neron model is unique. In 2), the isomorphisms you write are only true over the algebraic closure. Have a look at Chapter 10 of Liu's book for more "scheme-theoretic" proofs. The same chapter also answers 3). Indeed, split multiplicative is the same as split semi-stable (+ singular). I don't completely understand 4). If $E\to S$ is an elliptic curve (so flat in particular) over a scheme $S$, then for all $s$ in $S$ the fibre $E_s$ is an elliptic curve. This is the reduction of $E/S$ at $s$.
May
10
comment $R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.
@floflo You say that $f$ has smooth fibres. In particular, assuming $X$ is integral and $Y$ is a smooth curve, then $f$ is flat so that $f$ is a smooth morphism. Therefore $U=Y$ in your second paragraph. Also, a local system being trivial usually means it's zero. In any case, what you claim is not true if $n$ is odd. I tried explaining this in my answer below.
May
10
answered $R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.
Apr
28
comment Fibers of a scheme over $\text{Spec}\,\mathbb{Z}$
@miguels The affine line and the projective line over $\mathbb Z$ answered your first two questions. To construct schemes with less points just remove points from these schemes. For instance, you can remove the zero section of $\mathbb A^1_{\mathbb Z}$ to get a scheme with $p-1$ rational points in its fiber over $(p)$.
Apr
14
comment Surface constructed using curves
@dario Compute $E^2$ by writing it as $p^{-1}(y) \cdot p^{-1}(y)$ with $y$ some point in $F$ and $p$ the projection $E\times F\to F$.
Mar
26
comment Higher dimensional analogue for Riemann Hurwitz formula
The setting in the higher-dimensional case is similar to the case of curves: you start with a reasonable map $f:X\to Y$ (say proper with finite fibres) and you want to relate topological invariants of $X$ to topological invariants of $Y$. Unfortunately, one of the points made in the answers to the questions you link to is that you should forget about "numerical relations between topological invariants" and rather work with equalities of line bundles (or divisors). So maybe try to learn a bit about canonical bundles and understand the meaning of Riemann-Hurwitz: $K_X = f*K_Y + R$.