All Questions
Tagged with semigroups-and-monoids ag.algebraic-geometry
24 questions
9
votes
0
answers
347
views
What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 1,871
6
votes
2
answers
781
views
What is the combinatorial data classifying non-normal affine toric varieties?
Recall that a toric variety is a variety $V$ containing an open dense algebraic torus. Here an algebraic torus means a finite product of copies of the multiplicative group of the ground field (which I ...
- 38.6k
6
votes
0
answers
294
views
Laurent and power series over the field with one element?
Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For ...
user1437
5
votes
1
answer
359
views
Computations of divisor class monoids
Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
- 26k
5
votes
1
answer
303
views
A characterisation of faces of rational polyhedral cones
This is about a (seemingly) basic lemma about rational polyhedral cones that is sometimes used when working with toric varieties and is usually "left to the reader". Unfortunately, I could ...
- 6,700
4
votes
2
answers
364
views
General linear inverse monoid
Let $V$ be a finite dimensional vector space over some field (say, $\mathbb C$). Consider the set $\operatorname{GLI}(V)$ of all linear isomorphisms between subspaces of $V$. This is a monoid under ...
user6976
4
votes
0
answers
174
views
Centers and conjugacy classes of groups relative to a pair of group homomorphisms
$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by
\begin{align*}
\mathrm{Z}(G) &\...
- 11.8k
4
votes
0
answers
330
views
determine if a toric variety is Gorenstein
Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod End(V_{\omega_{i}})\times\prod\...
- 3,472
4
votes
0
answers
331
views
What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
3
votes
1
answer
313
views
Are local rings of monoid algebras geometrically unibranch?
Let $\mathrm{M}$ be a finitely generated submonoid of $\mathbb{Z}^{\oplus d}$ for some $d$, let $A := k[\mathrm{M}]$ be the associated monoid algebra over a field $k$, let $\mathfrak{m} \subset A$ be ...
- 2,017
3
votes
1
answer
189
views
Faces of polyhedral cones and open immersions of affine toric schemes
Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.
Let $\sigma\subseteq V$...
- 6,700
2
votes
1
answer
174
views
Understanding the picture of monoidal space
Ogus in his slides https://math.berkeley.edu/~ogus/preprints/colloqhandout.pdf presents the following picture of a monoidal space $\operatorname{Spec}(\mathbb{N} \longrightarrow \mathbb{C}[\mathbb{N}])...
- 165
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
2
votes
0
answers
73
views
What should I call a log scheme with free reduced monoids?
This is a terminology question about a class of log varieties.
Given an fs (fine and saturated) log variety $(X, M)$ (for $M$ the defining sheaf of monoids), any geometric point $x\in X$ has a ...
- 7,952
1
vote
2
answers
296
views
Are orbits of an affine algebraic monoid affine?
Let us work over the complex numbers for simplicity. Let $M$ be an affine algebraic monoid and $X$ an affine variety on which $M$ acts regularly, i.e. there is a morphism $\alpha: M\times X\to X$. Let ...
- 4,902
1
vote
1
answer
106
views
Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
1
vote
1
answer
134
views
amalgamated sum of monoids
Consider the amalgamated sum $Q_1 \rightarrow^{v_1} Q_1 \oplus_P Q_2 \leftarrow^{v_2} Q_2$ of $Q_1 \leftarrow^{u_1} P \rightarrow^{u_2} Q_2$ with $Q_1,Q_2,P$ being monoids.
Why does $v:= v_i \circ u_i$...
- 65
1
vote
0
answers
355
views
On logarithmic schemes
I have two questions on logarithmic schemes
Can we explicitly construct a chart for any coherent logarithmic scheme? By definition of coherence it must have a chart but given a coherent sheaf of ...
- 494
1
vote
0
answers
55
views
Schemes for conditional distributions (monads)
(Note: This is a heuristic question. I'm trying to work out if this idea makes sense. I don't have much background in this area, so apologies if I'm wide of the mark.)
Suppose you have a monad ...
- 121
1
vote
0
answers
111
views
When a semigroup ideal is a determinantal ideal?
Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \...
- 113
1
vote
0
answers
254
views
Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
- 65
1
vote
0
answers
143
views
on reductive monoids which are gorenstein
Let $M$ a reductive monoid, i.e. a integral normal affine scheme, which is a monoid whose group of units is a connected reductive group.
By Rittatore http://www.cmat.edu.uy/cmat/docentes/alvaro/...
- 3,472
0
votes
0
answers
114
views
Clarifications sought on the paper on the semigroup associated with a free polynomial by Ali Abbas and Abdallah Assi
I have three questions regarding the proof of Proposition 4 on page 4 of this paper here. For those interested in addressing these questions, please refer to some definitions in the first two or three ...
