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 ...
Will Sawin's user avatar
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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, $S_1$ ...
abx's user avatar
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8 votes
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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^*\...
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 ...
Sasha's user avatar
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6 votes
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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 $...
abx's user avatar
  • 37.1k
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. ...
ssx's user avatar
  • 2,729
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.
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 ...
Robert Bryant's user avatar
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 ...
Willie Wong's user avatar
  • 37.4k
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 ...
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 $...
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 ...
Mohammad Ghomi's user avatar
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.
Enrico's user avatar
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