Tagged Questions

1
vote
1answer
71 views

analytically isomorphic singularities

Two plane curves $X,Y$, defined by polynomials $f(x,y)=0$ and $g(x,y)=0$,are analytically isomorphic(at the origin). i.e., the complete local rings $k[[x,y]]/(f)$ and $k[[x,y]]/(g) …
4
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1answer
73 views

$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$

Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the shea …
6
votes
1answer
142 views

Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is cont …
6
votes
2answers
94 views

Integer lattice points on a hypersphere

Is the following statement true? For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ i …
1
vote
1answer
93 views

General Orthogonal Group and its properties

I know that exist a Lie Group Called the Orthogonal Group $O(n)$. That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for …
5
votes
1answer
148 views

Constructing Polynomial Count Varieties

I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$: Are all reductive algebraic groups strongly polynomial-count? Are products of strongly poly …
0
votes
1answer
65 views

common roots of bivariate polynomial equations

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but differ …
0
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1answer
35 views

Connection between the number of vertices and the number of lattice points of the integer hull of a polytope?

Is there a connections between the number of vertices and the number of lattice points of $P_I$, the integer hull of a polytope $P$? Which is usually more difficult to determine? O …
9
votes
1answer
256 views

Smith Normal Form of powers of a matrix

What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix? The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R …
0
votes
1answer
109 views

local complete intersection

The following is an Exercise 1.1.11 of Hartshorne's Algebraic Geometry. Let $Y\subset \mathbb{A}^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)$ is …
3
votes
0answers
89 views

mixed Hodge polynomial

Let $X$ be a smooth projective algebraic variety over a field of characteristic zero. Let $U$ be the complement in $X$ of a simple normal crossings divisor $D$. For each degree $k$ …
-2
votes
2answers
188 views

Embedded associated prime and non zero divisor

$M$ is a finitely generated $A$-module of dimension $d$ such that $G(M)$ is eqidimensional and $M$ does not have any embedded prime. Given $x\in I$ where $I$ is an ideal of $A$ an …
3
votes
3answers
263 views

Surfaces ruled over elliptic curves

Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve. Suppose $E$ …
17
votes
2answers
613 views

How can I randomly draw an ensemble of unit vectors that sum to zero?

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a …
4
votes
2answers
113 views

When is the intersection of an isolated normal singularity with a generic linear subspace through that singularity normal?

Suppose I have an affine subvariety $A \subset {\mathbb C}^N$ of dimension $n \geq 3$ which has an isolated singularity at $0$ (lets say for the sake of simplicity that it is non-s …

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