Questions tagged [hypersurfaces]
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33
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Is a variety always contained in a hypersurface of smaller or equal degree?
(a) Let $V\subset \mathbb{A}^n$ be an affine variety (not necessarily irreducible). Write $\deg(V)$ for the sum of the degrees of its irreducible components. Must there be a hypersurface $W\subset \...
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Are there CM complete intersections of arbitrarily large degree and codimension?
For every $d, c$ does there exist a complete intersection $X \subset \mathbb{P}^N$ of codimension $c$ and multidegrees $d_1, \dots, d_c \ge d$ such that the Mumford-Tate group of $X$ is abelian?
The ...
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Algebraic hypersurfaces and Coxeter groups
What is the minimum degree of an algebraic hypersurface (not necessarily smooth) having each Coxeter group as its symmetry group?
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Real (non-complex) Du Val singularities for quartics of high global Milnor number
I posted this in MathStackexchange and was advised to come here. As I am only looking for examples, I didn't feel MathOverflow was necessary.
I am looking for examples of specific quartic projective ...
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Tangent bundle of Milnor manifold
As I have been studying about Milnor manifold defined above, I want to understand its tangent bundle structure. I could not find anything related to that anywhere.
I am aware of the fact that $H(m,n)$ ...
1
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1
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Closed surfaces of prescribed mean curvature
Let $D\subset\mathbb R^n$ be a smoothly bounded open domain and $0\in D$. For any $x\in\partial D$ there holds
\begin{eqnarray*}
2 \,a'(\vert x\vert)\,(x\cdot\nu(x))+(n-1)\,a(\vert x\vert) \, H(x) = \...
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3
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Open complement of hypersurfaces
Let $k$ be an algebraically closed field. Let $H_1, H_2$ be two smooth hypersurfaces of the same degree $d$ in $P^n_k$. Let $U_1,U_2$ be their complements respectively. Are $U_1,U_2$ isomorphic as ...
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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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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 ...
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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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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?
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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 ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 \...
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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$,...
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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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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 &...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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