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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 twodimensional 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

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Direct image of structural sheaf
and for all coherent sheaves $\mathscr{F}$. 
1d

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Direct image of structural sheaf
Yes. See [Hartshorne, Algebraic Geometry], Corollary 11.2 page 279. 
2d

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Polinominal equations
Clean my living room 
Jan 29 
revised 
Fano manifold admit an smooth anticanonical divisor?
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Jan 29 
revised 
Fano manifold admit an smooth anticanonical divisor?
added 16 characters in body 
Jan 29 
answered  Fano manifold admit an smooth anticanonical 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 
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Proof of a cubic equation problem
The line $x+y=0, \, zw=0$ is contained in the Fermat projective surface $x^3+y^3+z^3w^3=0$, 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
added 494 characters in body 
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 
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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 
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Example of a fibration into singular curves?
If the varieties involved are smooth and we are in characteristic $0$, by BertiniSard's Lemma the general fibre must be smooth. 
Jan 10 
answered  Brouwer vs. Cantor 