10
votes
Open complement of hypersurfaces
If $U_1$ and $U_2$ are isomorphic then $H_1$ and $H_2$ are equal in the Grothendieck ring of varieties and thus, by the Larsen-Lunts theorem, stably birational, which if $d>n$ implies that they are ...
9
votes
Accepted
Open complement of hypersurfaces
The answer is no. Perhaps the simplest case is $n=2$, $d=4$. There is a unique double covering $\pi _i:S_i\rightarrow \mathbb{P}^2$ branched along $H_i$. If $U_1$ and $U_2$ are isomorphic, $S_1$ ...
8
votes
Accepted
Is a cubic hypersurface determined by its Fano variety of lines?
Edit. Regarding the Picard group, since $\rho^{-1}(F(X))$ is a $\mathbb{P}^1$-bundle over $F(X)$, the Picard group $\text{Pic}(\rho^{-1}(F(X)))$ equals $\text{Pic}(F(X))\oplus \mathbb{Z}\cdot [\pi^*\...
Community wiki
7
votes
Open complement of hypersurfaces
The easiest case is $n = 1$, $d = 4$. Indeed, the embeddings $U_i \to \mathbb{P}^1$ are canonical, hence an isomorphism $U_1 \cong U_2$ extends to an isomorphism of the ambient projective lines and ...
6
votes
Accepted
Birational morphism and invariance of arithmetic genus
These are two different questions.
1) No. The arithmetic genus of a degree $d$ surface $Y\subset\mathbb{P}^3$ is $\chi (\mathcal{O}_Y)-1=\binom{d-1}{3}$, regardless of the singularities of $Y$. If $...
5
votes
Is a cubic hypersurface determined by its Fano variety of lines?
For what it's worth, I think I now know a complete proof. I learned of this proof from D. Huybrechts's notes in progress http://www.math.uni-bonn.de/people/huybrech/Notes.pdf, proposition 6.21.
...
4
votes
Hypersurfaces whose equation is not known
The projective dual of a variety is usually a hypersurface. So, take your favorite variety and try to compute its projective dual.
Community wiki
4
votes
Accepted
Umbilic points on Euclidean hypersurfaces
Because people have asked for it, I thought I would supply an example of what I mentioned in my comment above, an immersion of the $3$-sphere into $\mathbb{R}^4$ that has three distinct principal ...
3
votes
Closed surfaces of prescribed mean curvature
If you let $\partial D$ be a hypersurface of revolution whose generatrix $\gamma: I \to [0,\infty) \times \mathbb{R}$ is such that $|\gamma|$ is monotonic, you then have $H(x)$ and $x\cdot \nu(x)$ can ...
3
votes
Irreducibility of the singular locus of a cubic hypersurface
Edit. I missed the condition that the secant variety should span the hypersurface. I am leaving the example below for any case. I will think about the secant condition.
Original Post.
That is not ...
Community wiki
2
votes
Accepted
Does a moving family of lines through a fixed point produce a singularity?
To expand the comment of @potentially dense: the answer is $r-2$, and it does not depend on the degree of $X$. In general, suppose $X\subset \mathbb{P}^r$ is a hypersurface, and $p$ a smooth point of $...
Community wiki
2
votes
Approximating a compact $C^1$ hypersurface without boundary
It seems that what you are asking is essentially how one can approximate the area of a surface via triangulations. This is a classical topic stretching back more than a century. The famous example of ...
1
vote
Are there algorithmic tools for computing poincare residues?
This answer of mine might help. To compute in concrete examples the Jacobian ring any computer algebra tool (e.g. Macaulay2) will do.
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