Questions tagged [hypersurfaces]

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0answers
85 views

Description of determinantal varieties in $\mathbb{P}^n$ that are linear sections of determinantal varieties in $\mathbb{P}^{n+1}$

Fix an algebraically closed field $k$ of characteristic 0. Consider an $n$-tuple $(A_1,\ldots, A_n)$ of $n\times n$ matrices over $k$ and assign to it the determinantal surface in $\mathbb{P}_k^{n-1}$ ...
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0answers
37 views

Lipschitz hypersurface

I asked this already on Math SE. Maybe this definition is not quite common, but I'm asking myself what a Lipschitz hypersurface is. Intuitively this is a hypersurface which can locally be parametrized ...
1
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0answers
42 views

Uniformly graphical hypersurfaces in Riemannian manifolds

Let $M$ be a hypersurface embedded in $\mathbb{R}^n$. It is known that if the norm squared of the second fundamental form of $M$ is bounded, then we can find a uniform lower bound for the radius $R>...
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0answers
139 views

Dimension of quintic hypersurfaces singular at given number of points

How many quintic hypersurfaces are there which are singular at given points (need not be general) of length at least 20? Is there any upper bound of the dimension of such quintics?
3
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0answers
86 views

Calculation of the mean curvature under a normal perturbation

Let $X: M^n \to N^{n+1}$ be a Riemannian immersion. Write $g, A, \nu, H$ for the first fundamental form, second fundamental form, Gauss map and mean curvature of $X$ respectively. Consider the normal ...
2
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1answer
199 views

Quadric surfaces tangent to a cubic threefold along a line of first type

Take a line $L$ of the first type on a smooth cubic threefold $X$ over $\mathbb C$, then its normal bundle $N_{L|X}$ is isomorphic to $\mathcal{O}_L\oplus \mathcal{O}_L$. This is equivalent to say ...
4
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0answers
111 views

Area lower bound given a mean curvature upper bound?

If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
0
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0answers
116 views

Local complete intersection and hypersurfaces

Let $Y \subset \mathbb{P}^n$ be a regular, codimension $2$, complete intersection subscheme in $\mathbb{P}^n$ (for example, $Y \cong \mathbb{P}^{n-2}$). Let $X$ be a normal (not necessarily smooth) ...
3
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0answers
135 views

Existence of regular hypersurface sections

Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...
2
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0answers
72 views

Is Colding-Minicozzi entropy continuous w.r.t. $C^\infty$ convergenge?

For an hypersurface $\Sigma^n \subseteq \mathbb{R}^{n+1}$ the entropy introduced by Colding and Minicozzi (see their paper) is defined as $$ \lambda(\Sigma) := \sup_{x_0 \in \mathbb{R}^{n+1} \\ t_0 \...
3
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0answers
129 views

Hilbert polynomial of structure sheaf of hypersurfaces

Is there an example of a hypersurface $X$ of some projective space $\mathbb{P}^n$ such that there exists an invertible sheaf $\mathcal{L}$ on $X$, not isomorphic to the structure sheaf $\mathcal{O}_X$,...
15
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2answers
704 views

Is a cubic hypersurface determined by its Fano variety of lines?

Consider a smooth cubic complex hypersurface $X\subset\mathbf{P}^{n+1}$ of dimension $n\geqslant 3$. The associated Fano variety of lines $F(X)$ is a smooth variety of dimension $2n-4$. Can one ...
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0answers
227 views

Are smooth specializations of smooth hypersurfaces again hypersurfaces

Let $X\subset \mathbb{P}^n$ be a smooth projective hypersurface of degree $d$ (over the complex numbers). Assume $n$ is very large compared to $d$, and that $d$ is a prime number (e.g., $d=3$ and $n &...
1
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1answer
185 views

Approximating a compact $C^1$ hypersurface without boundary

Can we approximate (arbitrarily closely) a compact $C^1$ hypersurface in Euclidean space without boundary with a polygonal hypersurface, such as a simplicial complex? To clarify, I want to have the $\...
2
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0answers
95 views

How do conformal maps affect curvature?

Let $(\overline{M}^{n+1}, \langle \cdot, \cdot \rangle)$ be a riemannian manifold with riemannian connection $\overline{\nabla}$ and consider $M^n \subset \overline{M}$ an orientable hypersurface with ...
6
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1answer
191 views

Hypersurfaces whose equation is not known

I would like to find some well-known/interesting hypersurfaces which arise as parametrizations where implicitization is computationally too difficult. I have software which computes the Newton ...
3
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1answer
195 views

Are there algorithmic tools for computing poincare residues?

In Schnell's note on Computing Picard-Fuchs Equations he gives a recursive method for computing residues on hypersurfaces. In short, if you have a meromorphic differential form $$ \frac{dw}{w^k}\wedge ...
4
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0answers
232 views

English language and Mathematics

I have a question maybe more relevant to an English language section of StackExchange, but I doubt that anybody but a Mathematician could properly answer my question. Let $\mathcal M$ be a smooth ...
1
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1answer
329 views

Birational morphism and invariance of arithmetic genus

Let $f:X \to Y$ be a birational morphism between projective, irreducible surfaces. Assume $X$ is non-singular and $Y$ is a hypersurface in $\mathbb{P}^3$ (not necessarily smooth). Is the arithmetic ...
2
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0answers
164 views

Computing higher dimensional intersection numbers for complete intersections of $\mathbb P^n$

Let $X_1,X_2$ be two smooth hypersurfaces of degree $d$ in $\mathbb P^{n}$. Let $B=X_1\cap X_2$. Assume $B$ is smooth. Let $\mathcal N_{B/\mathbb P^n}$ be the normal bundle to $B$. Let $H$ be the ...
1
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1answer
188 views

Does a moving family of lines through a fixed point produce a singularity?

This is just a feeling that I had and I am curious if it is totally wrong or true to some extent. Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In ...
4
votes
1answer
284 views

Umbilic points on Euclidean hypersurfaces

Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...
3
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1answer
514 views

Irreducibility of the singular locus of a cubic hypersurface

Let $Z\subseteq \mathbb{P}^{N}$ be an irreducible cubic hypersurface, i.e. $Z=V(F)$ for certain homogeneous irreducible polynomial $F\in K[X_{0},\ldots,X_{N}]$ of degree $3$. Let us suppose that its ...
0
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0answers
72 views

Obtaining Hessian of the embedding from an induced metric

Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric ...
2
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1answer
495 views

Number of singular fibers in families of hypersurfaces

Consider the projection map $$\pi: X = V(t_0 f + t_1 gh) \to \mathbf P^1,$$ where $[t_0: t_1]$ are the homogeneous coordinates on $\mathbf P^1$, $f=f(x_0, \dots, x_n)$ is a homogeneous polynomial of ...
23
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5answers
6k views

Intuition for mean curvature

I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form. Is there ...