Tagged Questions
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
2
votes
1answer
59 views
blow-up of $\mathbb{P}^5$ as a projective bundle
I am wondering if the blow-up of $\mathbb{P}^5$ along three disjoints $\mathbb{P}^1$ (say in generic position) can be understood as a projective bundle over some nice (Fano?) variety.
If one ...
-1
votes
1answer
66 views
extension of Riemannian metric on real affine variety
Given a Riemannian metric $g$ on the real part $X_R$ of a real affine variety $X$,
is there a "natural" way to extend $g$ to be a Riemannian metric on $X$?
1
vote
0answers
123 views
Find the Range of Function
What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$,
$$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$
where $b_k=\prod_{j\neq k}(z_j-z_k)$ for $1\leq k\leq m$ ?
10
votes
1answer
208 views
Why is the kth cohomology group of the DM-compactification of the moduli space of curves pure of weight k?
I'm trying to understand the paper
Arbarello, Enrico, Cornalba, Maurizio,
Calculating cohomology groups of moduli spaces of curves via algebraic geometry.
Inst. Hautes Études Sci. Publ. Math. No. 88 ...
0
votes
0answers
62 views
Profinite Local Ring inside Polynomial Ring
This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...
1
vote
2answers
314 views
Is it meaningful to work on convergencies, integration, etc. on the Zariski topology? [on hold]
Since I have studied analysis as well as algebra recently, I am familiar to work on integrablities, and such concepts when I look at topologies. Currently, I am studying algebraic geometry, and I want ...
0
votes
0answers
25 views
Depth of multigraded modules
Can any one please give me some references on depth of multigraded module over a standard multigraded ring.
2
votes
1answer
122 views
Does normalization of projective varieties preserve very ampleness
Let $f:\tilde{X} \to X$ be a normalization of projective variety. Let $L$ be a very ample line bundle on $X$. Is $f^*L$ a very ample line bundle on $\tilde{X}$? If not true in general, is there any ...
0
votes
0answers
92 views
Parabolic bundles on elliptic curves
as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
9
votes
0answers
160 views
Lifting Abelian Varieties to p-adic fields
Assume I have an abelian variety $A$ over a finite field $k$ of characteristic $p$. Work of Norman and Oort (1980) says I can lift $A$ to an abelian variety $\mathscr{A}$ over some characteristic ...
3
votes
1answer
131 views
Intersection theory on M_{g,n}
Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
-3
votes
0answers
84 views
Veronese surface [on hold]
I have a question(Hartshorne ,page 13,exercise 13):
If Y be the image of the 2-uple embedding(described in exercise 12) of P^2 in P^5.
and if Z(a subset of Y) is a closed curve(variety of dim 1) show ...
1
vote
1answer
116 views
Bott formula for projective bundles
For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
1
vote
1answer
108 views
Kaehler form on weighted projective space
The Kaehler potential for the standard Fubini-Study Kaehler form in projective space $\mathbb{C} P^n$ is given by:
$$\log(\sum_{i=0}^n |z_i|^2)).$$
What is the analogous formula for a Kaehler ...
2
votes
1answer
668 views
Can we construct cohomolgy theory on noetherian separated schemes without Axiom of Choice?
The usual cohomology theory on schemes uses injective or flasque resolutions of quasi-coherent sheaves. Hence it uses Axiom of Choice.
However, if the base scheme is a noetherian separated scheme, the ...
9
votes
2answers
314 views
What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$
The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...
3
votes
1answer
91 views
Simple example of isolated critical point with non-semisimple monodromy
Consider a polynomial map $f :\mathbb{C}^{n+1} \rightarrow \mathbb{C}$ with $f(0)=0$ (no constant term) and with isolated critical point at $0 \in \mathbb{C}^{n+1}$. We can choose a disc $D$ of some ...
5
votes
0answers
73 views
Comparison of K-groups of (affine) singular schemes with K'=G-groups
It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
9
votes
4answers
278 views
Is any quadric birational to a product of Brauer-Severi varieties?
Let $k$ be a field with algebraic closure $\bar k$. Assume that $k$ is perfect and not of characteristic $2$ for simplicity. Let
$$X: \quad Q(x)=0, \quad \subset \mathbb{P}^n_k,$$
be a non-singular ...
8
votes
2answers
286 views
The moduli space of special Lagrangian submanifolds
Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
4
votes
1answer
151 views
Blow-up of $\mathbb{P}^4$ along a quadric surface
Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
2
votes
0answers
49 views
Quasi-finite morphisms of stacks
Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite?
By a "nice" stack I mean a smooth finite ...
0
votes
0answers
48 views
plotting parametrized algebraic curves near singularities
I have a parametrized algebraic curve:
x(t)=A(t)/D(t);
y(t)=B(t)/D(t);
with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...
1
vote
0answers
64 views
Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy
Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and
$\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...
0
votes
0answers
82 views
Degree and quasi projective family
Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$.
Question : $\exists ...
2
votes
0answers
71 views
Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?
Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
13
votes
3answers
1k views
Algebra and Cancer Research
Let me start by acknowledging the existence of this thread: Mathematics and cancer research ?
It is well-known that mathematical modeling and computational biology are effective tools in cancer ...
4
votes
2answers
324 views
Holomorphic trivialization of $(x,y) \subset \mathbb{C}[x,y]/(y^2 - x^3 + x)$
This question is largely out of curiosity but also motivated by an attempt to understand vector bundles on elliptic curves better.
I believe it is a theorem of Grauert that any holomorphic vector ...
3
votes
0answers
238 views
What is the status of the Friedlander-Milnor conjecture today?
For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...
3
votes
1answer
182 views
Is the big cell a principal open set?
Let $G$ be a complex affine reductive algebraic group, $B\subseteq G$ a Borel with maximal torus $T$ and unipotent radical $U$. Let $w\in\operatorname N_G(T)$ be a representative of the longest Weyl ...
4
votes
1answer
204 views
Fermat surface known to have very few rational integer solutions
The motivation for this question is the Selmer curve, given by
$$\displaystyle 3x^3 + 4y^3 + 5z^3 = 0.$$
One can show that this curve has no rational integer solutions, despite having a solution ...
0
votes
0answers
50 views
Surjectivity of $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$
Let $R$ be a Noetherian ring and let $M$ is finitely generated $R$-module.Suppose $p$ is a minimal prime in $\text{Supp}_RM$. Then $f\colon M\rightarrow \Gamma_{pR_p}(M_p)$ that $f(m)=m /1 $ is ...
1
vote
1answer
144 views
Proof of the Belyi's theorem: where it is really used the hypothesis?
Consider the Belyi's theorem:
If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most ...
12
votes
2answers
311 views
Do most degree $d$ morphisms of $P^n$ have smooth critical locus?
Let $f=[F,G,H]:\mathbb{P}^2\to\mathbb{P}^2$ be a morphism of degree $d\ge2$.
The critical locus $C_f$ of $f$ is the zero-locus of the Jacobian
determinant:
$$
C_f = \left\{ [x,y,z]\in\mathbb{P}^2 ...
0
votes
0answers
65 views
Explicit calculation of module of derivations on noncommutative polynomial ring
Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...
7
votes
1answer
154 views
Hasse principle and Brauer-Manin obstruction for forms of large degree
The Hasse principle is perhaps an at-first naive generalization of the Chinese remainder theorem; that if a linear equation can be solved modulo $p$ for any prime $p$, then it can be solved in the ...
1
vote
0answers
87 views
Over which fields (of positive characteristic) is the Beilinson-Soulé vanishing conjecture known to hold?
Let $k$ be a field, and denote by $K_p(k)^{(n)}$ the weight $n$ eigenspace of the Adams operations on the $p$-th $K$-group of $k$.
The Beilinson-Soulé (BS) vanishing conjecture predicts that
$$
...
3
votes
1answer
146 views
flat descent for perverse sheaves
Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
0
votes
1answer
139 views
A covering lemma of Kawamata
In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is
Let $X$ be a non-singular projective variety, and $D$ be a divisor ...
2
votes
1answer
110 views
Quotient of product of curves
Let $C_1,C_2$ be smooth, projective curves of genera $g_1,g_2 \geq 2$. Assume that a group $G$ of order $(g_1 - 1)(g_2 - 1)$ acts on $C_1$ and $C_2$ such that $C_1/G \cong \mathbb{P}^1$ and $C_2/G ...
1
vote
1answer
95 views
Can the property of essential finite type checked at a point?
Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.
Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...
2
votes
1answer
200 views
representation of algebraic fundamental group of projective line minus three point
everyone, I want to ask is there any result in the literature
similar to the following:
Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X_{\mathbb{Q}}$ ...
2
votes
1answer
172 views
Canonical lifts from $\mathbb F_q$ and CM-theory
One knows that (ordinary) Jacobians of hyperelliptic curves over a finite field $\mathbb F_q$ (mostly of genus 1 (elliptic curves) and 2) are extensively studied by cryptographers, as a platform for ...
4
votes
6answers
938 views
Algebraic Geometry for non-mathematician [closed]
I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a ...
3
votes
0answers
216 views
Van den Bergh Duality, Serre Daulity and Poincaré duality [closed]
All three duality theorems:
Van den Bergh Duality, Serre Duality and Poincaré duality seem to be very similar, is there an explicit relationship between the three?
For example can van den Bergh ...
1
vote
0answers
149 views
Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$
Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...
1
vote
0answers
156 views
+50
Pullback of a sheaf associated to a divisor
I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...
7
votes
1answer
240 views
Top self-intersection of exceptional divisors
Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
3
votes
1answer
148 views
Smooth mixed hodge modules - representations of fundamental group?
I do not know much about mixed Hodge modules. I would like to ask: Let $X$ be a smooth connected algebraic complex variety, with a chosen point. Could one describe smooth mixed Hodge modules on $X$ as ...
1
vote
0answers
69 views
degree of isogenies between Jacobians and Abelian Varieties
Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...
