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12
votes
0answers
289 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 ...
7
votes
0answers
102 views

The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane

Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to ...
6
votes
0answers
126 views

Are Schubert varieties for Kac-Moody groups cut out by linear equations?

Let $G$ be a reductive group, and let $X$ be a partial flag variety for $G$. Then it is known that for any projective embedding of $X$, that the equations scheme-theoretically cutting out a Schubert ...
5
votes
0answers
189 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, ...
5
votes
0answers
262 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 ...
4
votes
0answers
280 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 ...
4
votes
0answers
646 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 ...
3
votes
0answers
87 views

Generalizing von Staudt's synthetic construction of the complex numbers

Starting from the real projective plane described synthetically/axiomatically, it is possible to construct the complex projective plane directly without passing through coordinates: one adjoins two ...
3
votes
0answers
122 views

Universal covering space of a Zariski open subset of projective space

Let $U$ be a Zariski open subset of $\mathbb P^n_{\mathbb C}$. Assume $U$ is the complement of some divisors. Have the possible universal covering spaces of $U$ been classified? Do we know when the ...
3
votes
0answers
120 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 $g$ ...
3
votes
0answers
311 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}= ...
3
votes
0answers
211 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 ...
2
votes
0answers
58 views

Resolution of indeterminacies for a map to Grassmannian of planes

Let $X\subset \mathbb P_k^N$ be a $n$-dimensional smooth projective variety ($n\geq 2$) and $\phi_l:X^l\dashrightarrow Gr(l,N+1)$ ($l\leq n+1$) be the natural rational map which associates to a ...
2
votes
0answers
53 views

Determine the representation given by space of sections of symmetric products of cotangent bundle of projective plane

In a recent project, it was interesting for me to determine the $PGL(3)$ representation given by $H^0(S^2(\Omega(1)) \otimes \mathcal O(4))$ on $\mathbb P^2$. I did this by using the Euler sequence, ...
2
votes
0answers
74 views

Rational curves through a fixed number of points

Let us fix two positive integers $d$, and $N$. Can we determine a third integer $n$ such that given $n$ general points $p_1,...,p_n\in\mathbb{P}^N$ there exists a unique rational curve of degree $d$ ...
2
votes
0answers
136 views

A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$. For instance consider a cubic surface ...
2
votes
0answers
168 views

Degree of join of two varieties‏

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...
2
votes
0answers
271 views

Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions: $f(ax) = af(x)$ for ...
2
votes
0answers
130 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 ...
2
votes
0answers
376 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, ...
2
votes
0answers
249 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 ...
2
votes
0answers
243 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 ...
1
vote
0answers
90 views

When is an intersection of quadrics empty

Is there an easy way to tell when the intersection of a set of quadrics (in complex projective space) is empty? Assume you know the quadrics explicitly.
1
vote
0answers
70 views

Intersection of affine subspaces with specific Veronese variety

Consider the mapping $$ \Psi: \mathbb R^2 \times \mathbb R^2 \to \mathbb R^5, \\ \Psi(x^{(1)},x^{(2)}) = \begin{pmatrix} x_1^{(1)} + x_1^{(2)} \\ x_2^{(1)} + x_2^{(2)} \\ (x_1^{(1)})^2 + ...
1
vote
0answers
219 views

Is there a projective metric on a projective space induced by a p-norm?

A projective metric on a projective space is a metric on the underlying set such that shortest path with respect to this metric are parts of entire projective straight lines. The 2-norm induces the ...
1
vote
0answers
72 views

Higher entry loci

Let $X\subset\mathbb{P}^n$ be a projective variety, and let $p\in\mathbb{P}^n$ be a general point. The secant cone $C_p(X)$ relative to $p$ is the union of all the secant line of $X$ through $p$. The ...
1
vote
0answers
149 views

A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...
1
vote
0answers
54 views

Projection from a polytope to an affine space

Let $P\subseteq \mathbf{R}^n$ be some polytope defined by an intersection of half spaces with corresponding hyperplanes $H_k$, and let $A\subseteq \mathbf{R}^n$ be some affine space, with $A\cap P ...
1
vote
0answers
212 views

Projective tangent cones, ordinary singularities and blow-ups

Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent: $X$ has an ...
1
vote
0answers
92 views

Reference for a proof of a projective representation of $A_6$

This question is copied from math.stackexchange, in hope that it might get some attension. I want to understand the proof of There is a projective representation of $A_6 \hookrightarrow ...
1
vote
0answers
129 views

Resolution of singularities of projective varieties

Let $X\subset\mathbb{P}^n$ be an irreducible variety, and let $Sing(X)$ be its singular locus. Let $Y$ be the blow-up of $\mathbb{P}^n$ along $Sing(X)$. Assume that we know that the strict transform ...
1
vote
0answers
206 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) ...
1
vote
0answers
157 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 ...
1
vote
0answers
174 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 ...
1
vote
0answers
89 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 ...
0
votes
0answers
51 views

Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
0
votes
0answers
88 views

Quadrics cutting out a polygon

Let $l_1,l_2,l_3,l_4,l_5\subset \mathbb P^4_k$ be distinct lines such that $|l_i\cap l_{i+1}|=1$ for all $i\ mod\ 5$ and $l_i\cap l_j\neq \emptyset\iff j=i+1\ mod\ 5$ (so that $\cup_{i=1}^5l_i$ is a ...
0
votes
0answers
35 views

cross sections of semialgebric sets

Let n be bigger than two, and let A be a subset of the n-dimensional euclidean space. Suppose that the intersection of A with any (n-1)-dimensional affine hyperplane is semialgebraic. Can one conclude ...
0
votes
0answers
127 views

Independent Generic Curves in the Projective Plane

I'm trying to read a paper by Masayoshi Nagata (available here) where he gives a counter-example to Hilbert's fourteenth prolem and I've run into some trouble understanding the terminology he's using. ...
0
votes
0answers
46 views

Pascal and Brianchon's theorems generalized for hyperbolic paraboloid

I know that giving a general version of these two theorems for quadrics can be quite tricky, but if we restrict ourselves to a verssion that holds for the hyperbolic paraboloid only it should be ...
0
votes
0answers
172 views

projective map from $\overline{\mathcal{M}}_{0,n}$

Suppose I have a morphism $f:\overline{\mathcal{M}}_{0,n} \to \mathbb{P}^N$ birational onto its image, and I know exactly what $F$-curves are contracted (or "dually", what divisors are contracted). ...
0
votes
0answers
106 views

Why is the set of geometrically irreducible curves of degree d in P^2 open in P^((d+2 2)-1)?

Let $X$ be a $k$-scheme. We say that $X$ is geometrically irreducible if $X\times_k \mathrm{spec}{K}$ is irreducible for all algebraically closed extensions $K$ of $k$. Ravi Vakil states in his ...
0
votes
0answers
174 views

classify antiholomorphic involutions of projective space

On $\mathbb{CP}^1$ with standard complex structure, how to prove that there are only two types of antiholomorphic involution, given by $$ \tau :[z:w]\mapsto [\bar w, \bar z] \qquad \eta :[z:w]\mapsto ...
0
votes
0answers
98 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. ...
0
votes
0answers
81 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 ...
0
votes
0answers
173 views

lefschetz theorem for quadrics

Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?
0
votes
0answers
246 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. ...
0
votes
0answers
202 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 ...