Francesco Polizzi
|
Registered User
|
|
|
9h |
revised |
Surfaces ruled over elliptic curves added 35 characters in body |
|
10h |
accepted | Surfaces ruled over elliptic curves |
|
11h |
revised |
Surfaces ruled over elliptic curves added 1107 characters in body; added 1 characters in body |
|
11h |
comment |
Surfaces ruled over elliptic curves @Will: thank you, this is precisely the part I was missing :-) |
|
1d |
answered | Surfaces ruled over elliptic curves |
|
2d |
accepted | Basics of minimal Elliptic Surfaces [following Beauville] |
|
2d |
revised |
Basics of minimal Elliptic Surfaces [following Beauville] added 31 characters in body |
|
2d |
revised |
Basics of minimal Elliptic Surfaces [following Beauville] added 386 characters in body |
|
2d |
answered | Basics of minimal Elliptic Surfaces [following Beauville] |
|
May 13 |
awarded | ● Enlightened |
|
May 13 |
awarded | ● Nice Answer |
|
May 13 |
comment |
Proving that a generic variety with ample canonical bundle has no automorphisms I have added some further details |
|
May 13 |
revised |
Proving that a generic variety with ample canonical bundle has no automorphisms added 611 characters in body; added 6 characters in body |
|
May 12 |
accepted | Proving that a generic variety with ample canonical bundle has no automorphisms |
|
May 12 |
comment |
Proving that a generic variety with ample canonical bundle has no automorphisms @Jonathan: 1. If you take the branch locus in a very ample linear system, by Bertini theorem the general element of the system will be smooth, hence the general cover will be smooth. 2. By the universal property of the Albanese map, the cover $\alpha \colon X \to A$ factors through the Albanese morphism $X \to \textrm{Alb}(X)$. Since $\deg \alpha =2$ and $X$ is of general type, the isogeny $\textrm{Alb}(X) \to A$ must be an isomorphism. 3. $X$ has non trivial deformations coming from the deformations of $A$ (and from those of the branch locus). |
|
May 12 |
comment |
Proving that a generic variety with ample canonical bundle has no automorphisms If you want a rigid (= without deformations) counterexample, maybe some fake projective plane could work. |
|
May 12 |
answered | Proving that a generic variety with ample canonical bundle has no automorphisms |
|
May 8 |
revised |
regularity of finite flat branched covers deleted 73 characters in body |
|
May 8 |
comment |
regularity of finite flat branched covers Alternatively, one can use Hurwitz formula to define the branch locus, at least when $C$ is Gorenstein. In fact, $K_C = f^*K_S + R$, where $R$ is the ramification divisor. Then $D=f_*R$ as a cycle, so it makes sense to speak of non-reduced branch divisor. For instance, in Joel example the ramification divisor is the intersection of the two lines. It is linearly equivalent to a general fibre of the covering, which is given by two distinct points sent by $f$ to the same point. So $f_*R=D$ is a point counted with multiplicity two. |
|
May 8 |
comment |
regularity of finite flat branched covers Ok, so let use the following definition that can be found in the paper by Iversen "Numerical invariants of multiple planes", p. 971. Let $x \in S$ and $y_1, \ldots, y_r$ be the points of $f^{-1}x$. Let $d_i$ be the discriminant of the extension $\widehat{\mathcal{O}_x} \to \widehat{\mathcal{O}_{y_i}}$. Then a local equation of $D$ in $x$ is given by $\prod d_i$. |
|
May 8 |
revised |
regularity of finite flat branched covers deleted 326 characters in body; added 14 characters in body; added 33 characters in body |
|
May 8 |
comment |
regularity of finite flat branched covers Joel: right. Anyway, in your example the schematic branch locus is not reduced (is one point counted with multiplicity two). Indeed, your cover is the limit of a family of flat double covers $f_t \colon \mathbb{P}^1 \to \mathbb{P}^1$, all branched in two points, when the two points come together. |
|
May 8 |
answered | regularity of finite flat branched covers |
|
May 3 |
revised |
Reference for the classification of (singular) degree 4 surfaces in $\mathbb{P}^3_{\mathbb{C}}$? edited title |
|
May 3 |
answered | Reference for the classification of (singular) degree 4 surfaces in $\mathbb{P}^3_{\mathbb{C}}$? |
|
May 3 |
comment |
The prime number $2$ Actually, because it is so even |
|
Apr 24 |
comment |
Simplest complex curves with isomorphic Jacobian You are welcome. Right, it is not clear what "simplest" should mean in this case. At any rate, in Howe's paper there are examples given by pairs of genus $2$ curves, which is of course the lowest possible genus. In this sense, such examples might be considered "simple". |
|
Apr 24 |
accepted | Simplest complex curves with isomorphic Jacobian |
|
Apr 24 |
answered | Simplest complex curves with isomorphic Jacobian |
|
Apr 22 |
comment |
Morphism with non-reduced special fibre I took the liberty to change the title. |
|
Apr 22 |
accepted | Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? |
|
Apr 21 |
revised |
Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? edited body |
|
Apr 21 |
revised |
Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? added 213 characters in body; added 10 characters in body; deleted 1 characters in body |
|
Apr 21 |
answered | Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? |
|
Apr 19 |
comment |
Equivalent definitions of ample bundles @Serge: the definition of Hartshorne, page 121 is clear: $E$ is generated by global sections if and only if it is a quotient of a free sheaf, i.e. a quotient of a direct sum of copies of $\mathcal{O}_X$. And $\mathcal{O}_X$ of course has this property! |
|
Apr 19 |
comment |
Equivalent definitions of ample bundles @Serge: If you take $E= \mathcal{O}_X$ and a point $x \in X$, from the exact sequence $$0 \to \mathcal{O}_X(-x) \to \mathcal{O}_X \to \mathcal{O}_x \to 0$$ you see immediately that, since $H^0(\mathcal{O}_X(-x))=0$, the map $H^0(\mathcal{O}_X) \to H^0(\mathcal{O}_x)$ is an isomorphism, so $(1)$ is satisfied. |
|
Apr 19 |
revised |
Equivalent definitions of ample bundles added 84 characters in body |
|
Apr 19 |
answered | Equivalent definitions of ample bundles |
|
Apr 17 |
answered | Albanese dual to the Picard scheme |
|
Apr 11 |
comment |
Are period domains ever contractible "half-space", "half-plane": what do you think? |
|
Apr 9 |
accepted | Minimal compactification |
|
Apr 9 |
revised |
Minimal compactification deleted 12 characters in body |
|
Apr 9 |
answered | Minimal compactification |
|
Apr 4 |
revised |
Linkage between singularities of algebraic varieties and continued fractions edited title |
|
Apr 4 |
comment |
Linkage between singularities of algebraic varieties and continued fractions You can have a look at Section 2 of my paper with E. Mistretta: arxiv.org/abs/0805.1424 where the case of dimension $2$ cyclic quotient singularities is considered. Sorry for being self-referential :-) |
|
Apr 3 |
comment |
What is the definition of a sufficiently ample line bundle? Maybe you should say where precisely in the book you found this concept... |
|
Apr 3 |
comment |
who invented projective space $\mathbb{P}^n$? I do not know. Anyway, it seems that homogeneous coordinates were introduced by Moebius en.wikipedia.org/wiki/Homogeneous_coordinates |
|
Apr 3 |
comment |
Existence of smooth surfaces containing a curve mmh...so maybe after all there is still a mistake in my argument. Probably it is the following: for any $p \in C$ the set given by the surfaces in $I_C(d)$ which are smooth at $p$ is an open (hence dense) subset $U_p \subseteq |I_C(d)|$. But then it is possible that $\bigcap_{p \in C} U_p = \emptyset$. Is it this what you are saying? |
|
Apr 3 |
comment |
Existence of smooth surfaces containing a curve @Olivier: it seems to me that the argument proving that the base locus of $I_C(d)$ consists of $C$ only (for $d >>0$) assumes that $C$ is considered with its reduced structure, and that it cannot be carried out if instead $C$ has some nilpotent structure (as Jason's example shows). I'm missing something? |
|
Apr 3 |
revised |
Existence of smooth surfaces containing a curve added 42 characters in body; added 9 characters in body |
