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226 views

### Embedding of the product of two Grassmannians into a Grassmannian

Consider an embedding $$\Phi: G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})\rightarrow G_k(R^n)$$ of the product of two Grassmannians $G_{k_1}(R^{n_1})\times G_{k_2}(R^{n_2})$ into $G_k(R^n)$, where ...

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141 views

### Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...

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230 views

### Dual of a weighted projective space

I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...

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233 views

### Quaternionic Veronese Embedding

I know that the complex projective line $\mathbb{C}P^1$ can be embedded in the complex projective space $\mathbb{C}P^n$ (Veronese embedding). For example, $\mathbb{C}P^1\rightarrow \mathbb{C}P^3$ is ...

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104 views

### On the Haar measure of Grassmanians

How can one write down the Haar measure of complex Grassmanians in terms of Plucker coordinates? Is there any way to define a Kahlarian measure like $d\mu\propto \det(g)dp_{ij}dp^{*}_{ij}$ where $g$ ...

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71 views

### What criteria are to determine if two projective varieties are projectively equivalent?

A projective transformation is a morphism of $P^n$ to $P^n$, for some $n$, determined by an
$(n + 1) \times (n + 1)$ invertible matrix $A$ in the obvious way. The sets $Q$, $R$ are projectively ...

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183 views

### Polarizations on $M_{0,n}$ from Kapranov's quotient constructions

In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These ...

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96 views

### semi-orthogonal decompositions and embeddings

This is most likely a stupid question, but I am curious. It is well known that $\mathbb{P}^n$ has a semi-orthogonal decomposition
$$D^b(X)=\langle \mathcal{O},\dots,\mathcal{O}(n)\rangle$$
Suppose ...

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296 views

### Partitioning the Projective Plane

Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$.
Side Note: If there are more standard definitions for any of the ideas presented here, ...

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212 views

### Global sections of a coherent sheaf in terms of a presentation

Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an ...

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323 views

### What is projective duality from modern point of view ? (correspondence ? Fourier on D-mod ? Aut of D(Coh) ?)

Consider vector space V and its dual V^* then to any line subspace in V one can correspond its kernel in V^* which is hyperplane.
Projective duality states that this correspondence satisfies many ...

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251 views

### Galois group decomposition of non-cyclic covers

If $\pi: C \rightarrow \mathbb{P}^{1}$ is a cyclic cover of $\mathbb{P}^{1}$ with Galois group $\mathbb{Z}/m \mathbb{Z}$ and thus with the (affine) formula
$y^{m}= ...

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225 views

### quasi-trigonal curves

I have read in the literature about quasi-trigonal curves. Such a curve C is a hyperelliptic curve X with two points p,q identified (basically a pinch). They seem to be pretty important in the theory ...

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146 views

### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

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135 views

### What is the Birkhoff norm of a Perron vector?

Let $A$ be a positive matrix. What is known about the Birkhoff norm of its Perron vector?
By the Birkhoff norm of a vector $x$ I refer to the quantity $\frac{\max{x}}{\min{x}}$.
P.S. This is ...

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87 views

### invariant lines avoiding fixed subvarieties

Could anybody help me with the following question ?
Assume we are given:
(1) a finite order (linear) automorphism $g$ of the complex projective space $\mathbb{P}^r$,
(2) a closed algebraic ...

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25 views

### Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function
$f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...

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72 views

### Embed the normalization of a curve in a larger space

I'd like to believe that this problem has a positive answer, but I don't know a nice reference. Actually I've never worked with embedded curves, so I apologize in advance if the question is too silly.
...

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166 views

### canonical model of a reducible curve

Let $C$ be a stable reducible curve. Is there a natural way to define it's canonical model (I guess via the dualizing sheaf)? And does somehow the dualizing sheaf restrict to the (probably twisted) ...

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72 views

### sphere with projective structure

In " Geometric structures on low-dimensional manifolds " , section 2 , we have :
given a projective tame 3-manifold with radial ends , each end surface has a projective structure since a developing ...

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159 views

### lefschetz theorem for quadrics

Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?

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186 views

### Projective spaces with nonconstant regular functions

I can construct a scheme by patching that represents a projective space over an arbitrary ring. I can also prove that, if the ring is a Jacobson domain, the only regular functions on it are constants.
...

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170 views

### Does the normalization of a projective morphism determine the line bundle?

Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms
$$f:X \to \mathbb{P}^n$$
and
$$g:X \to \mathbb{P}^m,$$
such that the image of $f$ is the ...

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910 views

### How to get homography matrix in such case? (from one still image)

In my case as input I have such data structures: original image (RGB pixels), objects (squares) with lines crossing points in pixels (x,y) on Image Plane.
I have image like this
In my particular ...