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Mathematician, researching in Algebraic Geometry and related topics.


1h
revised Smoothing transverse intersections
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1h
revised Smoothing transverse intersections
deleted 27 characters in body; edited tags
3h
revised Smoothing transverse intersections
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3h
answered Smoothing transverse intersections
1d
comment holomorphic curves invariant by lattices
If $n \geq 2$, there is in general no such a function. Indeed, the $\Lambda$-invariant curve $C$ would give rise to a compact $1$-dimensional submanifold in the complex torus $T = \mathbb{C}^n / \Lambda$, but it is known that for a general choice of the lattice such a torus contains no $1$-dimensional complex submanifolds at all. So, your lattice must be special. For instance, you can require that the torus $T$ is algebraic (i.e, an abelian variety), which means that $\Lambda$ satisfies the Riemann bilinear conditions.
Mar
23
comment Disjoint curves in an algebraic surface
@MarcoGolla: do you have a reference?
Mar
23
comment Disjoint curves in an algebraic surface
Donaldson's theorem concerns simply connected 4-manifolds. However, there are (non rational) complex surfaces with $p_g=q=0$ that are not simply connected (for instance, the classical Godeaux surface), so in this case I do not see how to conclude that the intersection form must be necessarily diagonalisable. I'm missing something?
Mar
20
comment Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
I'm far for being an expert so I'm not sure whether this makes sense, anyway did you try to build a counterexample by using some kind of plane-filling curve?
Mar
19
comment Relative tangent space to proper morphism and irreducibility of fibers
Some authors require irreducibility in the definition of variety (e.g, Hartshorne), but some others do not (e.g, Shafarevich), so it is better to specify.
Mar
19
comment Relative tangent space to proper morphism and irreducibility of fibers
So, are you assuming that $X_y$ is a connected, non-singular variety?
Mar
16
revised Singular models of K3 surfaces
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Mar
16
revised Singular models of K3 surfaces
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Mar
16
answered Singular models of K3 surfaces
Mar
13
comment Covering of schemes and flatness
The normalization of the cuspidal curve is actually bijective (in fact, it is a homeomorphism), but the fibre over the cusp is not reduced.
Mar
13
comment Analytically but not algebraically smoothable singularity
Right, thanks. Corrected (and reference added).
Mar
13
revised Analytically but not algebraically smoothable singularity
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Mar
13
answered Analytically but not algebraically smoothable singularity
Mar
2
awarded  Good Answer
Mar
2
revised Residual finiteness: why do we care?
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Feb
27
awarded  Nice Answer