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Mathematician, researching in Algebraic Geometry and related topics.
Apr 29 |
comment |
Determine existence of irreducible variety in given homology class
You are right, I was imprecise. If the divisor is base-point free and the linear system $|D|$ is not composed with a pencil (i.e., the image of the associate morphism $\phi_{|D|} \colon X \to \mathbb{P}^N$ has dimension $\geq 2$), then you can make it irreducible by Bertini. |
Apr 29 |
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Showing two Rings are nor isomorphic
An isomorphism between the two rings would give an isomorphism of the corresponding spectra. But a circle and a hyperbola are not isomorphic over $\mathbb{R}$: the former is connected but the latter is not. |
Apr 29 |
comment |
Determine existence of irreducible variety in given homology class
@Will: If the divisor is very ample (or more generally base-point-free) you can make it irreducible by Bertini. If the divisor is only ample, this is not in general true: there are examples of reducible, ample divisors which are rigid. |
Apr 28 |
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Pull-back of reflexive sheaves
True, I'm considering the simplest case. |
Apr 28 |
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Pull-back of reflexive sheaves
Yes. A reflexive sheaf on $X$ is locally free outside a closed subset $D$ of codimension $\geq 3$. Now, for instance, take $Y$ disjoint from $D$. |
Apr 28 |
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Pull-back of reflexive sheaves
Of course it is possible. For instance, if $\mathcal{F}$ is a vector bundle (=locally free sheaf), so is its pullback (=restriction to $Y$). |
Apr 28 |
awarded | Enlightened |
Apr 27 |
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Is there a nonzero sheaf with all cohomologies vanish?
Take an elliptic curve $E$ and a degree $0$ coherent sheaf of the form $\mathcal{O}_E(p-q)$, where $p$ and $q$ are two distinct point on $E$. |
Apr 27 |
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Synthetic projective definition of cubic curves
Uhm, not that I'm aware of. But I'm not really an expert of synthetic projective geometry. |
Apr 26 |
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A combinatorial question about orthonormal bases
What do you mean by "generate"? The abelian group $A$ is not fixed. For instance, if $f \colon S^{n-1} \to \mathbb{R}$ is any function satisfying your property, then we can construct another function $g \colon S^{n-1} \to S^1$ by setting $g(x) = e^{if(x)}$, where $S^1 \subset \mathbb{C}$ is seen as a subgroup of the abelian multiplicative group $\mathbb{C}^*$. |
Apr 22 |
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Is dimension invariant under blow-ups?
The dimension of an algebraic variety is a birational invariant (it equals the trascendental degree of the function field) so it is invariant under birational modifications, in particular invariant under blow-up. |
Apr 22 |
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Synthetic projective definition of cubic curves
I'm assuming that the ground field is any algebraically closed field $k$ of characteristic $0$ (for instance $k= \mathbb{C})$. The chapter on curves in Hartshorne's book is a good introduction for all this stuff. |
Apr 22 |
revised |
Synthetic projective definition of cubic curves
added 428 characters in body |
Apr 22 |
comment |
Synthetic projective definition of cubic curves
It is a $1$-dimensional linear system of degree $2$ over a curve. By Riemann-Roch, since a smooth plane curve has genus $1$, every effective divisor of degree $2$ defines a $g^1_2$ on it. The classical term for such a object was "rational involution": in fact, every $g^1_2$ determines uniquely a degree $2$ morphism $C \to \mathbb{P}^1$. |
Apr 22 |
revised |
Synthetic projective definition of cubic curves
edited tags |
Apr 22 |
answered | Synthetic projective definition of cubic curves |
Apr 22 |
awarded | Nice Answer |
Apr 21 |
awarded | Enlightened |
Apr 21 |
awarded | Nice Answer |
Apr 21 |
answered | Must an algebraic variety with trivial tangent bundle be an abelian variety? |