Francesco Polizzi
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 37m comment Picard groups of quartic K3 surfaces It is possible to construct a K3 surface $S$ with Picard rank $2$ by taking a double cover of the plane of the form $w^2=F_6(x, \, y, \, z)$, where $F_6$ is a smooth plane sextic curve which admits a six-tangent conic. The surface $S$ does not sit naturally in $\mathbb{P}^4$, but in the weighted projective space $\mathbb{P}(1, \, 1, \, 1, \, 3)$. See Example 20 here: www-fourier.ujf-grenoble.fr/sites/ifmaquette.ujf-grenoble.fr/… 18h answered Picard groups of quartic K3 surfaces 1d comment Great Mathematicians Without a PhD Starting from when? Otherwise, I could say Apollonius and Archimedes. Anyway, I do not think this question is suitable for MO. Apr 29 comment Use of algebraic topology in geometry(differential and complex analytic) This is an extremely broad question. There are hundred of situations where algebraic topology is useful for the study of geometry: monodromy of coverings, classifying spaces, characteristic classes, invariants of singularities, etc etc. It is really impossible to cover such a plenty of applications in a MathOverflow answer. Voting to close. Apr 29 comment Early examples of problems that are easier in high dimension "but here the 3D result does readily imply the 2D one" Yes, but only for some projective planes, namely the ones where you can perform this lifting construction to the $3$-space. In fact, Desargues theorem is valid in all projective spaces of dimension $\geq 3$, but there are some non-desarguesian projective planes. So the case of dimension $2$ is really the tricky part here. Is not this precisely an example of what you are looking for? Apr 29 answered Which varieties are flat degenerations of projective space? Apr 28 awarded Nice Answer Apr 28 revised Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why? added 34 characters in body Apr 28 answered General Reference for surface singularities Apr 27 comment Can the matrix exponential be equal to the elementwise exponential oops, right :-) Apr 27 comment Can the matrix exponential be equal to the elementwise exponential How does your argument rule out a $2\times 2$ diagonal matrix? Apr 27 comment Question regarding Geometric meaning of Noether normalization theorem for projective varieties Have a look at Emerton's answer in this thread: mathoverflow.net/questions/42275/… Apr 22 comment Calabi-Yau with nodes A hypersurface singularity is always Gorenstein, and it is normal as soon as it is of codimension at least $2$. Apr 22 revised Chern classes of ideal sheaf of an analytic subset edited body Apr 21 revised Equivalence of Branched Coverings deleted 3 characters in body Apr 21 answered “Smooth” rationally connectivity Apr 20 comment Self-intersection of a Cartier divisor Yes, under your assumptions the movable part $M$ of $|D|$ is necessarily composed with a base-point free rational pencil, hence $M^2=0$. Apr 20 revised Self-intersection of a Cartier divisor added 6 characters in body; edited title Apr 20 revised Self-intersection of a Cartier divisor added 35 characters in body Apr 20 revised Self-intersection of a Cartier divisor added 35 characters in body