Questions tagged [ag.algebraic-geometry]
for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
14,774 questions
2
votes
1answer
103 views
Divisibility of a divisor
Let $X$ be a smooth complex projective curve and $f \colon X \to Y$ an étale Galois cover, whose Galois group $G$ is finite and of order $r$. For any $g \in G$, define $$\Delta_g = \{(x, \, g \cdot x) ...
3
votes
0answers
95 views
Functoriality of base change morphisms
Consider a commutative diagram of morphisms of schemes:
$$\begin{array}{ccccccccc}
X_2 & \xrightarrow{j_2} & Y_2 \\
f'\downarrow & & \downarrow f \\
X_1 & \xrightarrow{j_1} &...
2
votes
0answers
84 views
Geometric irreducibility of fiber product of geometric irreducible schemes
Given three geometrically irreducible normal $k$-curves, $X$, $Y$, $Z$, and two morphisms $f \colon\ X \to Z$, $g \colon\ Y \to Z$. Assume that $X \times_Z Y$ is irreducible. Does $X \times_Z Y$ is ...
1
vote
0answers
112 views
Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
1
vote
0answers
74 views
The moduli scheme of “$\nu$-canonically embeded curves”
This is related to the proposition 5.1. of Mumford's GIT.
It states that:
There is a unique subscheme $H$ in the Hilbert scheme $Hilb_{\mathbb{P}^n}^{P(x)}$ such that, for any morphism $f : S \to ...
2
votes
0answers
132 views
De Rham cohomology and extension of scalars
Let $K$ be a field of characteristic zero and let $X$ be a smooth variety over $K$. Given a field extension $L/K$, I denote by $X_L$ the variety $X \times_{Spec(K)} Spec(L)$.
What is the easiest way ...
7
votes
1answer
212 views
Embeddings of flag manifolds
Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
-1
votes
1answer
209 views
Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$
My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...
2
votes
0answers
86 views
Extending section of étale morphism of adic spaces
This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...
7
votes
1answer
155 views
Equivalence of definitions of Cohen-Macaulay type
I know that the Cohen-Macaulay type has this two definitions:
Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring; $M$ a finite $R$-module of depth t. The number $r(M) = dim_k Ext_R^...
2
votes
0answers
176 views
The moduli scheme of smooth curves of given genus is irreducible
I've heard that Deligne-Mumford's "the irreducibility..." showed this first.
But I think that Mumford's "Geometric invariant theory" has its proof.
The proof is as follows:
Let $H$ be the scheme that ...
3
votes
0answers
86 views
Automorphy Factor from Vector Bundles on Compact Dual
So I'm coming from an algebraic geometry perspective and I'm trying to carefully piece together the story of interpreting automorphic forms as sections of vector bundles on Shimura varieties. I think ...
0
votes
0answers
76 views
Reducible section of algebraic varieties
Let $X$ be a smooth algebraic variety and $Y$ a reducible linear section of $X$, then is it true (or under which assumptions it is true) that $Y$ is contained in a reducible hyperplane section of $X$ (...
12
votes
0answers
273 views
When is a map of topological spaces homotopy equivalent to an algebraic map?
My question is simple, but I don't expect there are any simple answers.
Let $X$ and $Y$ be a pair of schemes, and let $X(\mathbb{C})$ and $Y(\mathbb{C})$ denote their respective spaces complex points....
-2
votes
0answers
148 views
Injective map on spectra [on hold]
I know that if we have a surjection $f:B\rightarrow A$, this induces an injection on the spectra $f^* \colon \operatorname{Spec} A\hookrightarrow \operatorname{Spec} B.$
What about the opposite? ...
5
votes
0answers
143 views
Atiyah class and coboundary map
Let $L$ be a line bundle on a smooth algebraic variety $X$. Let $\sigma_i:U_i \times \mathbb{C} \to L_{|U_i} $
be its local trivializaations and $u_{ij}$ the transition functions satisfying $\sigma_j=...
1
vote
0answers
92 views
Core of the Jordan quiver variety
It is known that, given the Jordan quiver, dimension vectors
$\textbf{v}=n,\textbf{w}=1$ and a stability condition $\theta<0,$ the corresponding quiver variety $\mathcal{M}_{\theta}(n,1)\cong \...
1
vote
0answers
140 views
Level structures in deformation spaces of $p$-divisible groups
I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition ...
0
votes
0answers
89 views
When is geometric irreducibility an intrinsic property?
Let $k$ be a field. Let's say that $k$ has property $(∗)$ iff for a scheme $X$ and morphisms $f_1,f_2:X\rightarrow \mathrm{Spec}\,k$, geometric irreducibility of $f_2$ is implied by geometric ...
1
vote
0answers
95 views
Density of $k$-rational points for non-algebraically closed field
Let $k$ be a perfect field.
We say that $k$ has property $(*)$ if for every geometrically irreducible morphism locally of finite type $X\rightarrow \mathrm{Spec}\, k$ admitting a section, the set of ...
2
votes
0answers
69 views
Universal Property of the Zariski-Riemann Space
Let $k = \mathbb{Q}$ or $\mathbb{C}$. Let $K$ be a finitely generated field extension of $k$. A model of $K$ is a variety $V \subset \mathbb{CP}^n$ defined over $k$, such that the rational function ...
4
votes
0answers
99 views
The Zariski Riemann Space, but with Local Rings
The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...
3
votes
0answers
116 views
Is there a $\sum e_if_i=n$ in higher dimensions?
If $X\to Y$ is a finite map of connected proper algebraic curves over a field, then for any point $y\in Y$, the sum $\sum e_xf_x=n$ of ramification times inertia degrees over points $x$ mapping to $y$ ...
3
votes
0answers
75 views
Global functions on a flat proper family
Let $R$ be an integral domain. Let $f:X\rightarrow \mathrm{Spec}\,R$ be a flat proper morphism of schemes. Is it possible that $O_X(X)$ is not a flat $R$-module?
4
votes
1answer
137 views
Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?
There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
5
votes
1answer
124 views
Orbits of action of the split group of type $F_4$
Let the split group of type $F_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F_4\times \mathbb{G}_m$ on $A$, given by letting $\mathbb{G}_m$ act by scalar ...
3
votes
0answers
50 views
Metric with singularities on Riemann Surfaces and the associated Laplacians
I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...
0
votes
0answers
55 views
The algebraic variety behind a quotient of the ring of regular functions on the complex torus [closed]
I am trying to understand what algebraic variety stands behind the following quotient.
Let $I$ be the vanishing ideal of some linear subspace $\mathbb{C}^k \subset \mathbb{C}^n$ in the standard ...
2
votes
0answers
233 views
Topological data of $K3\times T^{2}$
I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold .
EDIT:...
6
votes
0answers
157 views
+50
On the algebraic cobordism cohomology of Eilenberg-MacLane spectrum
Let $MU$ be the complex cobordism spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum.
In this question it is shown that the group $MU^{*}(H\mathbb{Z})$ is non trivial.
From here, what can ...
1
vote
0answers
75 views
Norm quadrics and their motives
Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
6
votes
1answer
382 views
Poincare duality on the level of complexes
The classical Poincare duality is formulated in terms of cohomology groups. I am wondering if we can also formulate it in terms of complexes.
In particular, suppose $\mathcal{C}^*$ is a complex of $...
3
votes
2answers
238 views
Variety of conjugacy classes
Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
7
votes
0answers
220 views
Visualization and new geometry in higher stacks (soft question)
I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...
7
votes
1answer
130 views
The relative dimension of blow-up and singularities
Let $X$ be an integral affine scheme of finite type over $\mathbb{C}$, $\mathrm{dim}\,X=d$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get ...
2
votes
1answer
188 views
Étale fibration for $K[[X_1,…,X_n]]$
Let us consider a formal power series ring $A_n \colon= K[[X_1,\ldots,X_n]]$ with $0 \ll n < \infty$ and we shall consider a prime ideal ${\frak P}$ of $A_n$ such that $1 < {\mathrm{ht}}({\frak ...
3
votes
0answers
94 views
Computing affine Springer fibers
I'm having some trouble computing affine Springer fibers, even in simple cases. For example, consider the group $G=SL_2$ over $\mathbb{C}$ and let $\mathcal{K}=\mathbb{C}((z))$ and $\mathcal{O}=\...
-1
votes
0answers
205 views
A Riemann Surface can be characterized by its Lie Algebra? [closed]
I was studying the geometric properties of Lie algebras.
Let $X$, $Y$ be two Riemann surfaces. Suppose that this surfaces meet a finite set of ordinary differential equations. Let $L_X$ and $L_Y$ ...
2
votes
1answer
141 views
Semi-orthogonal decompositions for Calabi-Yau varieties
In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has ...
4
votes
0answers
180 views
Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$
I have a curious question about an argument/hint given in following thread:
https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes
The OP asked if ...
2
votes
0answers
81 views
Factoring Finite Morphisms through Open Immersions
This is a somewhat silly question, but I was wondering the following. Assume X, Y, and Z are schemes and consider the composition
$X \xrightarrow{f} Y \xrightarrow{g} Z $ and assume $g\circ f $ is a ...
2
votes
0answers
150 views
Property of Complete Variety
I have a question about a step in the proof from Lang's "Abelian Varieties" (page 20):
By definition an abelian variety $A$ over field $k$ is a proper smooth $k$-group scheme that is irreducible.
...
1
vote
1answer
103 views
Projective subvarieties of blow-ups of affine varieties
Let $X$ be an integral affine scheme of finite type over a field $k$. Let $Y\subset X$ be an integral closed subscheme of codimension $n>0$. We blow up $X$ along $Y$ and get a $k$-scheme $X'$. Is ...
8
votes
1answer
325 views
Why is, for a group scheme of finite type, “smooth” (resp. irreducible) equivalent to “geometrically reduced” (resp. geometrically irreducible)?
I have some questions about two statements from Bosch's "Algebraic Geometry and Commutative Algebra" about algebraic varieties (page 479). Since I still don't have the permission to add images I quote ...
6
votes
0answers
97 views
Is it possible for the Witt group of a scheme to have non-trivial odd torsion?
Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...
7
votes
3answers
415 views
Borel's presentation for the cohomology of a Flag Variety
If $G$ is a simple complex Lie group, $T\subset B\subset G$ is a choice of Borel and maximal torus, and $W$ is the Weyl group, then
1) $H^{*}(G/B,K)=K[T^{\vee}]/(K[T^\vee]^W_{+})$
and
2) $K[T^\vee]^...
4
votes
1answer
125 views
Automorphisms of singular hypersurfaces
Let $X\subset\mathbb{P}^{n+1}$ be an irreducible and reduced hypersurface of degree $d$.
A theorem by Matsumura and Monski asserts that if $n\geq 2$, $d\geq 3$, $(n,d)\neq (2,4)$ and $X$ is smooth ...
3
votes
0answers
152 views
Shimura varieties of Hodge type
I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...
9
votes
2answers
318 views
Nakano vanishing in positive characteristic
Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:
$(\ast) \quad$ $\mathrm H^...
4
votes
1answer
373 views
Do higher etale homotopy groups of spectrum of a field always vanish?
Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $\mathrm{Spec}\,k$ vanish?
If the absolute Galois group is finite, we have a universal cover $\mathrm{Spec}\,k^...
