# Questions tagged [hypersurfaces]

The hypersurfaces tag has no usage guidance.

26
questions

**5**

votes

**0**answers

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 $\mathbb{P}_k^{n-1}$ ...

**1**

vote

**0**answers

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**

vote

**0**answers

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>...

**0**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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 ...

**7**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**5**answers

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 ...