Skip to main content

All Questions

Filter by
Sorted by
Tagged with
3 votes
1 answer
232 views

Non-degeneracy in hyperplane intersections of canonical curves

Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
-4 votes
2 answers
6k views

Factorizing polynomials of several variables (in a different perespective)

I am looking for factorization of polynomials of several variables in the way outlined below. Consider a second degree polynomial of two variables over the complex numbers. "P(x,y) = Ax^2 + Bxy + Cy^...
3 votes
1 answer
156 views

How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?

Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$. I would like to show that $$ \{ \mathbf{x} \in \mathbb{A}^n_{\...
3 votes
0 answers
55 views

What is the precise definition of a quadratic form of Minkowski type (in the infinite case)?

I've been trying to understand a construction in the paper "Degree Growth of Meromorphic Surface Maps" by Bouksom, Favre and Jonsson. In it they state, In fact, the completion can be characterized ...
11 votes
1 answer
731 views

Reference request: Volume 2 of Abhyankar's lectures on algebra?

Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra. The description: This book is a timely survey of much of the algebra developed ...
4 votes
3 answers
382 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
2 votes
1 answer
217 views

Diagonalising a symmetric matrix with polynomial entries

Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
6 votes
1 answer
456 views

How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
16 votes
4 answers
3k views

How many minors I need to check to conclude all minors will vanish ?

Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish. I remember hearing that one really does not need to check all possible minors in order to conclude ...
7 votes
1 answer
727 views

Reference for Tate vector spaces

... aka locally linear compact vector spaces. The one reference I know is http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov3-10(CentExt).pdf. Does anyone know another good reference?