All Questions
8,187 questions with no upvoted or accepted answers
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426
views
The quotient of a scheme by a proper equivalence relation
Let $X$ be a scheme and $R$ be a proper equivalence relation on $X$.
What can be said about the geometric structure of the quotient $X/R$? Is it representable by a stack, for example?
- 3,472
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answers
130
views
morphisms in the construction of the moduli space of curves by mumford
Hi fellow mathematicians,
I'm just studying mumfords proof of the existence of a coarse modulispace for curves of genus $g$in his great GIT book. On page 102 (in the proof of proposition 5.2) he ...
- 9
0
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0
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440
views
non proper intersection
Let X and Y two smooth closed subschemes of a smooth projective scheme Z over a field.
Let $W:=X\cap Y$.
I suppose that W is non empty and that the intersection of X and Y is non proper, i.e
codim(...
- 3,472
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votes
0
answers
222
views
morphisms between algebraic spaces
My question concerns morphisms between algebraic spaces. I like the definitions of Artin, but I do not see a simple proof of the fact that the composition of two morphisms is a morphism. Could someone ...
- 7,722
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0
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164
views
Complex manifold $X$ with $\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2$
Are there any complex surface or threefold $X$ with
$$
\dim H^0(X,\Omega^{\dim_{\mathbb{C}} X})>2?
$$
I am asking this because I don't know any example while there are complex curves of genus ...
- 1,663
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0
answers
216
views
Applying Euclidean division Algorithm for polynomial in algebra software
The practical question is:
$P_5(w) = c_0 + c_1w + c_2w^2 +...+ c_5w^5$, where $ c_0, ... , c_5 $ are positive integers that I want to determine.
$Q_5(w) = w^2 (\frac{d}{dw}( \frac{P_5(w)}{w}) = -c_0 ...
- 11
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votes
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answers
467
views
Numerical dimension of nef divisors
Let $D$ be a nef divisor (moreover suppose it is effective if you prefer) on a normal projective variety of dimension $n$. Let $k\in[1,n-1]$.
If $D^k\cdot V=0$ for generic subvarieties $V\subseteq X$ ...
- 1,150
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195
views
lefschetz theorem for quadrics
Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?
- 3,779
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votes
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189
views
$T^2$-fibered K3 surface with involution
Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
- 1
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144
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From abstract to concrete complex projective varieties
By definition, a(n abstract) complex projective variety $X$ has an expression $E$ (many actually) of the form $\bigcap_i V_i$, with each $V_i$ a hypersurface in a ${\Bbb P}^m$ ($m$ depending on $E$). ...
- 17.6k
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164
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Maximum number of generators of a curve in $\mathbb{P}^3$
Let $H_{d,g}$ denote the Hilbert scheme of curves of degree $d$ and genus $g$ locally of complete intersection in $\mathbb{P}^3$. Given a curve $C \in H_{d,g}$, denote by $S(C)$ a minimal set of ...
- 1,070
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341
views
Computing fibre products of schemes (not affine)
It is clear how to compute the fiber product of two affine schemes. But Rhow can one compute the fiber product $X \times _{R} Spec(k[y])$ where $R$ is an integral domain, $X$ defined by the zero set ...
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597
views
what does it mean for a differential to be regular at a singular point?
Let $\omega$ be a differential form on a singular integral curve $X'$ over some algebraically closed field $k$ (ie, $\omega$ is an element of the stalk of the sheaf of differentials $\Omega_{X'}$ of $...
- 10.7k
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539
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.
...
- 133
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answers
152
views
Kählerdifferentials and normal crossing divisors
Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$
has normal crossings ...
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answers
171
views
Weaker conditions for potential good reduction of Abelian varieties
We are concerned with slight weakening of a result of the Serre-Tate paper "Good reduction of abelian varieties" google. I think the title of the question conveys what we are in here for. So, I'll ...
- 1
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answers
229
views
Reduction of endomorphism ring of Non-CM elliptic curve
Let $E$ be an elliptic curve defined over a number field without complex multiplication and with ordinary reduction at a prime $p\in\mathbb{N}$. When is the reduction mod $p$ map a surjection on the ...
- 1
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0
answers
359
views
Chern class of line bundle inducing a principal polarisation
What can one say about the Chern class $c_1(\mathcal{L})$ of a line bundle $\mathcal{L}$ on an Abelian variety $A$ inducing a (principal) polarisation $A \to A^\vee$?
Why am I asking this? My ...
user19475
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0
answers
394
views
finite homological dimension
Hi, I found the following in the proof of a theorem:
$ Z \subset Y \times M$ where $M$ is a smooth projective variety over $\mathbb{C}$, $Y$ is a scheme and $Z$ is a subscheme of the product, flat ...
- 73
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answers
100
views
Is this set of curves discrete?
Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 ...
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131
views
Do "recoil/rebound" left mutations exist (on a Del Pezzo surface)?
I have been studying exceptional sheaves and their mutations on Del Pezzo surfaces (specifically, on $\mathbb{P}^1 \times \mathbb{P}^1$). Given an exceptional pair $(E,F)$ of sheaves there are a ...
- 1
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answers
159
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Further Questions Regarding Cohomolgoy Theory Of Sheaves
In continuation to my previous post:
Question Regarding Riemann-Hurwitz Formula Proof
I'll be glad to receive some explanations regarding the following:
1) I know that when taking a sheaf $F$ , then ...
- 305
0
votes
0
answers
243
views
strict henselian and excellent henselian
Hello, everyone. I want to ask a problem about strict henselian ring.
Let $A$ be a strict henselian DVR.
Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are ...
- 1,921
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answers
101
views
Conditions under which the degree of an algebraic surface equals the degree of its planar section
An algebraic surface $\Phi$ in complex projective 3-space contains a circle $c$ such that the complex projective plane $P$ of $c$ intersects $\Phi$ only at the points of $c$. Assume that
$c$ is not a ...
- 1,488
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250
views
Intersections with divisors on moduli of curves
Let $\overline{\mathcal{M}}_{g,n}$ be the moduli stack of stable curves of genus $g$ with $n$ points.
Consider
$0 \neq \gamma \in Pic(\overline{\mathcal{M}}_{g,n})$
the first Chern class of a ...
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0
answers
131
views
Equalizers for morphisms of connected varieties with marked points
I recently began to study some aspects of varieties with marked points, and I tried to understand their categorical collective behaviors.
Here is one issue that I was not quite able to see ...
- 937
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236
views
On vanishing orders of an ideal via the restriction
Let $Y$ be a submanifold of a complex manifold $X$, and $a$ be an ideal on $X$ which does not vanish along the entire $Y$. Consider a point $\xi$ on $Y$, there are the vanishing order $ord_{\xi}a$ ...
- 320
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140
views
Obstruction theories on non-smooth spaces with smooth fibres
Given a perfect obstruction theory $E^\bullet$ over a space $X$, we know that if $X$ is smooth, that the virtual fundamental class $[X, E^\bullet]$ is given by
$$[X, E^\bullet] = c_{top}\big((E^{-1})^...
- 6,290
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votes
0
answers
86
views
Non log-canonical Fano of Picard number 1
What is an example of a Fano variety of Picard number 1 that does not have log-canonical singularities?
- 1
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votes
0
answers
746
views
Igusa model of modular curves
I would like to know what the "Igusa model" of the modular curve is and basic properties about it.
Can someone point me to a reference?
- 1,271
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254
views
comparison of cohomologies, Satake equivalence
If we regard the comparison isomorphism of cohomologies , we are given an isomorphism of Langlands dual group into itself; i wondered if it had any interest whatsoever, i cannot figure myself.
for ...
- 9
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0
answers
82
views
Extending functions on curves to functions on abelian varieties
Say I have a function $f_g$ on the moduli space of curves $M_g$ for all $g\geq 1$. Is there some way of extending this to the moduli space of abelian varieties $A_g$ in a nice way?
What if I have ...
- 163
0
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answers
303
views
Automorphism group of algebraic function fields
Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$?
Is it possible to do so if we know that $|G|$ is ...
- 2,179
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0
answers
115
views
the number of fixed points in geometric correspondance
Let $k$ be a finete field, $\bar{k}$ is an algebraic clousre of $k$, $\sigma \in Gal(\bar{k}/k)$ is the geometric Frobenius. Let $f:Y \to Spec(k)$ be a smooth, separated $k$-scheme of finite type, $...
- 1
0
votes
0
answers
169
views
Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
0
votes
0
answers
223
views
Can a surface of the following type contain a line?
This is a follow-up question to my previous post: Sufficient conditions to tell whether a surface contains a line
Suppose that $f(x_1, x_2)$ is an irreducible binary form with integer coefficients of ...
- 26.9k
0
votes
0
answers
282
views
well known facts on openness condition
Hi,
I would like to understand and prove the following two "well-known" facts:
1)
If $B$ is a scheme and $P$ a property for which I know:
i) if $B=Spec(V)$ where $V$ is a complete DVR, if $P$ ...
- 1
0
votes
0
answers
217
views
References needed for representation theory of certain unipotent algebraic groups in characteristic zero
Due to my preoccupation with characteristic p > 0 representation theory of algebraic groups, I am quite ignorant of some basic facts pertaining to their characteristic zero theory, mostly concerning ...
- 511
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0
answers
325
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Ordered Cech(-like) complexes that compute etale cohomology (of fields!)
It is well known (cf. Equivalence of ordered and unordered cech cohomology. ) that for 'usual' topologies one can compute the cohomology of sheaves either using unordered Cech complexes or ordered ...
- 16.9k
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186
views
Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?
Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...
- 16.9k
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answers
331
views
Properties of morphisms induced by divisors on curves
There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim |D|...
- 419
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0
answers
268
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
- 225
0
votes
0
answers
212
views
On 'special properties' of various 'sheaf image' functors for a local complete intersection morphism
Let $f:X\to Y$ be a local complete intersection morphism (of schemes or varieties) of (relative) dimension $c$ everywhere. Is it true that $f^!\cong f^*[2c]$ (as a functor between the derived ...
- 16.9k
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votes
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251
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 ...
- 1
0
votes
0
answers
159
views
a question on the Poincar\'e bundle
Let $C$, a smooth curve. Let $J$ its Jacobian, consider the Poincar\'e bundle $\mathcal{P}$ on $J\times J$. Let $p: J\times J\rightarrow J$ the projection.
How can I compute the complex $R p_{*} \...
0
votes
0
answers
240
views
Orbits of Infinite Grassmannian
"Any $L^+G$-orbit in Gr is known to be an orbit of the form Oλ for some λ ∈ $X_*(T)$, and Oλ = Oμ iff λ and μ are conjugate by W and the orbits form a finite stratification of each of the $Gr_i$."
...
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votes
0
answers
770
views
relation between pull-back of Cartier divisors, invertible sheaves and global sections
Let $\iota:X \hookrightarrow \mathbb{P}^n_k$ be a closed embedding of an irreducible non-singular projective $k$-scheme. For any hyperplane $H$ of $\mathbb{P}^n_k$ which doesn't contain $X$, since $X$ ...
- 1,109
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votes
0
answers
1k
views
push-forward of the structure sheaf, stein factorization, birational and connected fibers
I am confused with the following observation: Let $f : X:=\mathrm{Spec}(K) \rightarrow \mathrm{Spec}(k) =: Y$ be a scheme morphism corresponding to a non-trivial finite field extension ( hence $f$ is ...
- 1,109
0
votes
0
answers
148
views
Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization
Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.
Let $f:X\longrightarrow Y$ be a minimal resolution of ...
0
votes
0
answers
1k
views
Closed points on a variety
Hi,
I have a general question concerning the closed points $X^{0}$ of a variety $X$ over a field $k$:
one always hears that the properties of $X^{0}$ are "equivalent" to those of $X$, because it is ...
- 623