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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
comment 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
comment Pull-back of reflexive sheaves
True, I'm considering the simplest case.
Apr
28
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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?