Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,607
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Are spin Hurwitz numbers $r$-spin Hurwitz numbers?
(I think the answer is no, but I'm not sure.)
In Hurwitz theory, one counts $n$-fold branched covers $\Sigma'\to\Sigma$ of a Riemann surface $\Sigma$ with fixed
ramification data around each branch ...
- 6,766
3
votes
0
answers
227
views
Regular holonomic $\mathcal{D}$-modules
An holonomic $\mathcal{D}_X$-module on a smooth algebraic variety over $\mathbb{C}$ is called regular if all its composition factors are minimal extensions of the form $L(Y,N)$, where $N$ is a regular ...
- 1,315
2
votes
1
answer
450
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1-dimensional p-divisible groups, level structures and Cartier divisors
I am confused about the 1-dimensionality of $p$-divisible groups and its role in defining level structures.
Here's how I view/understand/not understand things:
If a $p$-divisible group arises from a ...
- 371
8
votes
1
answer
317
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Which ideals have standard Hilbert series?
Let $m$ and $d$ be two positive integers.
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...
CommunityBot
- 1
2
votes
2
answers
816
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Shimura datum of family of fake elliptic curves
Suppose we have a PEL type $(H,\phi ,*;T,O,V)$ where H is a rational nonsplit quaternion algebra, $\phi$is an embedding of Q-algebra $\phi : H-->M(2,R)$, and * is a positive anti involution of H; ...
- 3,674
12
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1
answer
2k
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Differential algebraic geometry vs Diffiety theory
Algebraic geometry is said to be useful to study not only specific solutions of polynomial equations but to understand the intrinsic properties of the totality of solutions of a system of equations.
...
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2
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1
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192
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When is the moduli of generalized parabolic bundles with fixed determinant smooth?
Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic ...
- 17.3k
6
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1
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669
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Rationality of the moduli space of genus g curves
I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
- 6,414
1
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1
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106
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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 ...
- 13.5k
8
votes
2
answers
631
views
Moduli 'space' of stacks?
In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow ...
- 42.5k
3
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1
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993
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How to calculate the Chern class of the tensor product of a torsion free sheaf with a line bundle
I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{...
- 37.3k
-1
votes
1
answer
800
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Kähler form on complex projective algebraic variety [closed]
I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...
CommunityBot
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1
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0
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106
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Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field
My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...
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1
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646
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The étale topos of a scheme is the classifying topos of which groupoid?
[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...
- 61.6k
9
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172
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Examples of curves with A7 symmetries
I'm trying to find examples of smooth algebraic curves with a (nontrivial) action of the alternating group $A_7$. Any examples are appreciated, and if there are classical examples, e.g. of genus 31 ...
- 1,138
2
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0
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94
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Class group of a 3-dimensional hypersurface singularity
For $f(z,w) \in \mathbb{C}[z,w]$ a square-free polynomial, consider an affine threefold
$$
X = V(xy + f(z,w)) \subset \mathbb{A}^4.
$$
By computing derivatives one sees that the singularities of $X$ ...
- 2,260
5
votes
2
answers
636
views
Embedding of a complex submanifold in projective space
Suppose you have a projective manifold $M$, a very ample bundle $\scr L$ and a transverse holomorphic section $s \in H^0(\scr L)$. Then the zero set of $s$ is a complex submanifold $S_M$.
Can we ...
- 949
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1
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233
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Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?
Nakajima & Yoshioka [1] showed that
\begin{equation}
F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
- 2,831
3
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2
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677
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Criteria for a coherent sheaf pushing forward from the universal cover
I would like to prove (or find a counterexample to) the following statement:
Let $X$ be a complex analytic scheme and let $\pi: Y \to X$ be its universal cover. Let $F$ be a coherent sheaf on $X$ and ...
- 137k
4
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0
answers
682
views
Kodaira's Fields medal citation
In 1954, Kodaira won the Fields medal. His citation was
Achieved major results in the theory of harmonic integrals and numerous applications to Kählerian and more specifically to algebraic ...
- 137k
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0
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294
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Deformation Vs. Blow up
Consider a surface $X$ (which is a projective variety over $\mathbb{C}$) with a double point singularity. To make $X$ smooth, we can either blow up the singularity with a $(-2)$ - curve ($\hat{X}$), ...
4
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73
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Identification of spectral and differential data for integrable difference equations?
Let $X$ be a projective curve and $G$ be a semisimple Lie group. There is a theorem roughly stating that there exists an isomorphism between the moduli space of principal $G$-bundles on $X$ and the ...
CommunityBot
- 1
1
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0
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180
views
Weaker version of smooth base change for étale sheaves
Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...
4
votes
1
answer
241
views
Enumerating the unfoldings of real singularities
Let $f: \mathbb{R}^n \to \mathbb{R}$ be a germ of an isolated, real analytic singularity. Let $I$ be the ideal in $R = \mathbb{R}[x_1, \dots, x_n]$ generated by the components of $\nabla f$, and $Q = ...
- 17.3k
7
votes
2
answers
945
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Explicitly describing the region of the plane "outward of" a simple, open, oriented, cubic curve $c:(0,1)\to\mathbb{R}^2$
Some Context:
I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an ...
- 701
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0
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130
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What can be said about $\{\deg(f),\deg(g),\deg(h)\}$, such that $k[f,g,h]=k[t]$?
Let $f=f(t),g=g(t),h=h(t) \in k[t]$, $k$ is a field of characteristic zero.
Denote $\deg(f)=a$, $\deg(g)=b$, $\deg(h)=c$. Assume that $a \geq 2, b \geq 2, c \geq 2$.
Assume that $k[f,g] \neq k[t]$, $...
- 2,783
3
votes
3
answers
335
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Intersection of a line and a conic in a surface
Does there exists a smooth projective surface $X$ which contains a projective line $L$ and a smooth conic $C$ such that $L\cap C=$empty?
- 37.3k
8
votes
1
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388
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projective plane cubics with exactly 9 real points
It is not hard to construct such curves explicitly, e.g. my favourite example is a curve $U$ singular at $(3:4:5i)$ and also passing through $(1:0:0)$, $(0:1:0)$, $(0:0:1)$, $(1:1:0)$, $(1:0:1)$, $(0:...
- 13.7k
1
vote
1
answer
680
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Restriction of vector bundles
I am trying to compute the Chern classes of the restriction of a rank two vector bundle on $\mathbb{P}^3$, denoted by $E$, with fixed Chern classes, $c_1(E) = c_1$ and $c_2(E) = c_2$, to a hyperplane $...
- 37.3k
5
votes
0
answers
196
views
When a coherent sheaf on DM stack is locally free?
A coherent sheaf $\mathcal{F}$ on a variety $X$ is locally free if every fiber $\mathcal{F}|_x$ is of the same dimension. My question is if such theorem is also true on a Deligne-Mumford stack, or ...
- 1,447
3
votes
0
answers
151
views
exact sequence of fundamental groups associated to "almost" smooth families of curves
Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not ...
- 1,308
6
votes
1
answer
740
views
Are there non-projective, but algebraic, hyperkahler varieties?
Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.
...
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1
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1
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68
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Example of a certain partitioned manifold
I'm looking for an example of a non-compact spin manifold $M$ and a compact subset $K\subseteq M$ such that $\partial K$ is a compact hypersurface in $M$ with $\hat{A}(\partial K)\neq 0$.
(At first I ...
- 18.8k
1
vote
1
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908
views
Hironaka's theorem and smooth completion
Hironaka's theorem states that for any algebraic variety (analytic space) $X$ there exists a smooth variety (complex manifold) $X'$ and a morphism $f : X' \rightarrow X$ such that $f$ restricted to $X ...
- 1,072
3
votes
0
answers
154
views
rational homotopy type and cohomological vanishing for local systems
Let $X$ be a (smooth) algebraic variety of complex dimension $n$, if for each local system $L$ with rational coefficients, we have
the vanishing $H^{i}(X,L)=0$, $\forall i>k$ for some fixed $k\geq ...
- 1,072
3
votes
1
answer
223
views
Isogeny of Drinfeld module
Let $\phi$ be a Drinfeld module over a function field $K$. It is known that if the rank of $\phi$ is 1, then it is isomorphic over $K$ to one defined over $O_K$. Is it true that if the rank of $\phi$ ...
- 1,958
2
votes
1
answer
741
views
isomorphism of line bundles over $\mathrm{Spec} \mathbb{Z}$
Suppose $X$ is a scheme, the structure morphism $X \rightarrow \mathrm{Spec}\mathbb{Z}$ smooth and surjective. Assume further that $H^0(X \times \mathrm{Spec} \mathbb{C}, \mathcal{O}^\times) = \mathbb{...
- 6,141
2
votes
0
answers
40
views
Asymptotic flag in terms of geometry of the stratum of abelian differentials?
Let $C$ be a closed Riemann surface of genus $g\geq 1$. Fix a holomorphic 1-form on $C$; it endows $C$ with a flat structure (i.e. a metric of trivial holonomy which has conical singularities at a ...
CommunityBot
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194
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Semisimplicity of the p-adic étale Tate module over $F_p(t)$
Let $k$ be a finitely generated field of positive characteristic p.
Let $A$ be an abelian variety over $k$ and write $T_p(A)$ for the $p$-adic étale Tate module of $A$. Is it known if the natural ...
- 718
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0
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210
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Curves in a non-normal surface
We are studying the behavior of families of curves inside stable families of surfaces. The non-existance of the following configurations of curves in a non-normal surface would be sufficient to prove ...
- 379
2
votes
1
answer
287
views
Lüroth theorem for $k \subset k(f,g) \subseteq k(x)$
Let $k$ be a field of characteristic zero (I do not mind to assume that $k=\mathbb{C}$, if things are easier in this case).
Lüroth theorem says that a field $L$, $k \subset L \subset k(x)$ containing ...
- 6,197
2
votes
0
answers
78
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Singularities of quotient of a vector bundle by a lattice
Let $V$ be a complex vector bundle of rank $g$ over an open unit disc $\Delta$ and $H$ be a integral local system of rank $2g$ over $\Delta$ i.e., for every $t \in \Delta$, $H_t \cong \mathbb{Z}^{2g}$....
- 1,573
7
votes
1
answer
923
views
Map Lifting lemma and Etale fundamental group
In algebraic topology, we have the map lifting lemma which says that given a covering space $p:(\tilde{X},\tilde{x})\rightarrow (X,x)$ and a map $f:(Y,y)\rightarrow (X,x)$ with $Y$ connected and ...
- 10k
12
votes
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402
views
Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?
Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor:
$$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
- 51.6k
4
votes
0
answers
290
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Derived categories of coherent sheaves and degenerations of abelian varieties
By the work of Burban-Drozd (https://projecteuclid.org/euclid.dmj/1076621984), we know what happens to the derived category of coherent sheaves when an elliptic curve degenerates into a nodal curve or ...
- 3,157
1
vote
0
answers
121
views
Sum of coefficients of a principal divisor
Let $X$ be an arithemetic surface. Let $(f)$ be a principal divisor. Let $D_{i}$ be the horizontal divisors showed up in $(f)$ and $d_i$ are their coefficients. I am trying to prove $\sum d_{i} m_i=0$,...
- 91
2
votes
0
answers
172
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Degeneration of a metric
I want to understand how the metric degenerate on a family of projective varieties (mainly for abelian varieties.).
Let $X$ be a smooth projective variety over $\mathbf{C}$.
Let $B$ be a smooth ...
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2
votes
2
answers
641
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When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?
Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$
be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the ...
- 3,235
5
votes
3
answers
845
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Poincaré duality for Deligne (co)homology
My question is about the papers
Hélène Esnault and Eckart Viehweg, Deligne-Beı̆linson cohomology
Uwe Jannsen, Deligne homology, Hodge-D-conjecture, and motives
(both from the Beı̆linson's ...
- 13.5k
5
votes
1
answer
141
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If $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$ equals $\bigwedge^k B$ for some complex matrix $B$, does it have a real source?
Let $1<k<d$ be an integer. Let $A \in \text{End}(\bigwedge^k \mathbb{R}^d)$, and suppose that $A=\bigwedge^k B$ for some complex $B \in \text{End}(\mathbb{C}^d)$.
Does there exist $M \in \...
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