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Mathematician, researching in Algebraic Geometry and related topics.


1d
comment projective map from $\overline{\mathcal{M}}_{0,n}$
In general the knowledge of the dual graph of the singularity does not identify its analytic type. When this happens, the singularity is called "taut". Taut two-dimensional singularities were classified by Laufer, see Math. Ann.205 (for instance, quotient singularities are taut). I do not know whether there are similar results in higher dimension.
1d
comment Direct image of structural sheaf
and for all coherent sheaves $\mathscr{F}$.
1d
comment Direct image of structural sheaf
Yes. See [Hartshorne, Algebraic Geometry], Corollary 11.2 page 279.
2d
comment Polinominal equations
Clean my living room
Jan
29
revised Fano manifold admit an smooth anti-canonical divisor?
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Jan
29
revised Fano manifold admit an smooth anti-canonical divisor?
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Jan
29
answered Fano manifold admit an smooth anti-canonical divisor?
Jan
28
answered When does a hyperelliptic Riemann surface admit a map of degree 3
Jan
23
comment Exercise from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris
I'm afraid MO is not the right place to ask about exercises on standard textbooks.
Jan
22
revised Isomorphism between a mapping class group and the fundamental group of a moduli space
edited tags
Jan
22
reviewed Approve Proof of a cubic equation problem
Jan
22
comment Proof of a cubic equation problem
The line $x+y=0, \, z-w=0$ is contained in the Fermat projective surface $x^3+y^3+z^3-w^3=0$, so you trivially have infinitely many integer solutions of the form $(x, \, -x, \, z, \, z)$.
Jan
21
revised Quintic polynomials generating cyclic extensions
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Jan
21
revised Quintic polynomials generating cyclic extensions
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Jan
21
revised Quintic polynomials generating cyclic extensions
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Jan
21
comment Quintic polynomials generating cyclic extensions
Then you should have added this in the background of your question :-) Anyway, I added a further reference.
Jan
21
answered Quintic polynomials generating cyclic extensions
Jan
16
comment Example of a fibration into singular curves?
If the total space is not smooth, you can have whatever you want. Take $X= C \times \mathbb{P}^1$ where $C$ is a nodal curve, and consider the first projection $f \colon X \to \mathbb{P}^1$. Then any fibre is isomorphic to $C$, hence singular.
Jan
16
comment Example of a fibration into singular curves?
If the varieties involved are smooth and we are in characteristic $0$, by Bertini-Sard's Lemma the general fibre must be smooth.
Jan
10
answered Brouwer vs. Cantor