Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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0
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236 views

Sheaf isomorphism $\mathcal{F}\rightarrow f_{\ast}f^{\ast}(\mathcal{F})$?

Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f_{\ast}f^{\ast}(\mathcal{F})$$ and localizing ...
5
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0answers
108 views

Algebraic fundamental group without regularity at infinity

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...
40
votes
2answers
1k views

What are reasons to believe that e is not a period?

I hope this rather soft question is suitable for MO, otherwise please migrate it to MSE. In their paper defining periods [1], Kontsevich and Zagier without further comment state that $e$ is ...
4
votes
1answer
176 views

Jacobian matrix and dimension of variety

I have 2 questions about the dimension of a complex variety: 1) Suppose I have a variety $V$ in $\mathbb{C}^n$ and an ideal $I$ such that $V=V(I)$ ($I$ may not be radical, i.e., it may differ from ...
1
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0answers
212 views

Feynman integrals in algebraic geometry [closed]

In quantum field theory, multi-loop Feynman integrals are basic ingredients of calculating high order corrections. Recently, I have come across the paper A Feynman integral via higher normal ...
1
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0answers
73 views

reference for Levelt-Turritin

Can anybody recommend a good reference to learn the Level-Turritin decomposition theorem of formal connections? An intuitive description of what it says would also be very appreciated.
2
votes
1answer
220 views

Is each rationally chain connected surface rational?

Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true): ...
0
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0answers
161 views

integral curves and differential equations on arcs

I am trying to prove a statement that is obivious in analytic setting, but makes me feel at a loss in formal algebraic setting. Let $M$ be a smooth curve over an algebraically closed field $k$. Let ...
1
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0answers
47 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 ...
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0answers
132 views

excess intersection theory

Can the excess intersection theory be applied to the following problem: I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, ...
2
votes
1answer
293 views

Paper by Mumford

In the paper of "The spectrum of difference operators and algebraic curves", by P. van Moerbeke and D. Mumford, Acta Mathematics, vol.143, 1979, (link: ...
2
votes
1answer
171 views

Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form $ \left( ...
10
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1answer
643 views

Reference request: Book of topology from “Topos” point of view

Question: Is there any book of topology in the modern language of topos theory? Motivation: In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
1
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0answers
107 views

Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...
6
votes
0answers
206 views

Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point. ...
5
votes
1answer
218 views

Realizing algebraic curves as complete intersections

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$). Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in ...
9
votes
1answer
535 views

Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...
1
vote
1answer
222 views

varieties whose canonical bundle has finite order in Pic?

Is there a structure theorem for such varieties? If X is a smooth and proper/projective variety whose canonical bundle $\omega_X$ has finite order in the Picard group, do we know anything about X? ...
5
votes
1answer
234 views

Existence of affine hulls

(This question is inspired by Matthieu Romagny's answer to my previous question about base change properties of affine hulls.) Given a scheme $S$, it is well-known (cf. EGA I.9.1.21) that the ...
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0answers
71 views

Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action $$ f: X_i \to (1+X_i)^{m} - 1 $$ on $C[[X_1,...,X_N]]$, where $C$ is the complex number field. Consider the analytic manifold $V(I)$ defined by the ...
1
vote
0answers
91 views

DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...
4
votes
2answers
380 views

Are linear algebraic groups rigid?

The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's ...
0
votes
1answer
216 views

irreducible etale cover of a blowup

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$. a.Does there exist a collection of smooth morphisms ...
5
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0answers
261 views

Counterexamples to Elkik's theorem in the non-Noetherian case

Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that: Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$. That is, if ...
3
votes
1answer
257 views

Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
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0answers
99 views

Galois descent of stacks

Suppose $X$ is a algebraic stack over a field $F$. Consider a Galois finite extension $F'$ of $F$. Let $X_{F'}:=X\times_F F'$ be the base extension. Suppose $X_{F'}$ is isomorphic to an algebraic ...
0
votes
1answer
122 views

linear section of codimension $k+1$ of a variety of dimension $k$

Let $X$ be a variety of dimension $k$ and degree $d$. If $L$ is a linear subspace of codimension $k+1$ such that $|L\cap X|$ consists of a finite number of points, is there a way to find the maximum ...
0
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0answers
48 views

(semi)stability of a net of quadrics and a certain criterion

Let $V:=H^0(\mathbb{P}^4,\mathcal{O}(1))$, $W:=H^0(\mathbb{P}^4,\mathcal{O}(2))$ and $Q_i(i=1,2,3)$ be quadric hypersurfaces in $\mathbb{P}^4$. To a net of quadrics $\Lambda=(Q_1,Q_2,Q_3)$, we ...
0
votes
1answer
113 views

Degree of a Segre variety (without Hilbert polynomial)

Algebraic Geometry is not my topic of research and I am having troubles in understanding the following: In Harris book, Algebraic Geometry, a firs course, Example 18.15, there is a proof of the ...
0
votes
1answer
67 views

Global sections in negative degree line bundles over singular curves

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
8
votes
1answer
799 views

What is the arithmetic Nullstellensatz?

The only precise statement (coming from a reliable source) of the "arithmetic Nullstellensatz" I can find is in Gowers's book, stating that two polynomials with integral coefficients have the same ...
-3
votes
3answers
105 views

if V(f) is irreducible, then how to show that the polynomial f itself is irreducible? [closed]

V(f) is the zero locus of the polynomial f in the polynomial ring k[x1, x2, ..., xn] with k an algebraically closed field. If V(f) is irreducible, then how to show that 'f' is irreducible?
0
votes
2answers
205 views

Weil restriction

I've already asked a similar question in SE, without success, so I've decided to post here a more general version of my question. Let $f: Y \to X$ be a finite etale morphism of smooth proper ...
1
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0answers
51 views

symmetric theta structures and arithmetic subgroups

A symmetric theta structure is a theta structure that commutes with (a lift of) the natural involution $\imath: A \to A$ an an abelian variety. For simplicity I will assume that $A$ is a surface. ...
4
votes
2answers
121 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
6
votes
2answers
325 views

Twist in identification with singular cohomology

Let $X$ be a smooth projective variety over $\mathbb{Q}$ and $$V = H^m(X(\mathbb{C}), \mathbb{Q} \cdot (2\pi i)^r)$$ Then I've seen people write the comparison with complex cohomology (an isomorphism ...
0
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0answers
151 views

Hermitian metric on $f_*(\Omega_{X/X_{can}}^{n-k})$

Let $X$ be an n- dimensional algebraic manifold . Suppose that its canonical line bundle $K_X$ is semi-positive and $0<k=Kod(X)<n $ . Let $f: X\to X_{can}\subset \mathbb CP^N$ Here $X_{can}$ ...
6
votes
2answers
322 views

Rational points techniques on curves not using their Jacobian

Let $C/K$ be a curve of genus > 2 over a number field $K$ and suppose there exists a $p \in C(K)$. Then a recurring theme in studying $C(K)$ is using the map $C \to J(C)$ normalized by sending $p$ to ...
2
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0answers
113 views

A question on resolution of singularities

I am wondering if it could be possible in particular cases to resolve a singularity of dimension $n$ by blowing-up a locus of dimension smaller than $n$. For instance consider a cubic surface ...
3
votes
2answers
90 views

Real solutions for systems of monomial equations

I have a $K$ equations of the form $x_1^{a_{i1}} \cdots x_n^{a_{in}}=c_i$ where $a_{ij}$ are non-negative integer constants and $c_i$ are real constants -- i.e. each equation is a monomial in $n$ ...
3
votes
0answers
107 views

Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...
5
votes
1answer
298 views

What is the current state of the crystalline analogue of the Weil conjectures?

In "F-isocrystals on open varieties results and conjectures" Faltings says: "Finally, we extend the theory of weights and show as much as possible of the crystalline analogue of the Weil ...
0
votes
0answers
182 views

Which of the Mochizuki's works are the most closely related to elliptic curves?

I'm very much interest about algebraic geometry and number theory along with cryptography, but I have a special interest about the elliptic curves. I have heard a lot of interesting things about ...
21
votes
2answers
935 views

A question about “Zariski dense” arguments

This question is a little basic, but I think it is consistent with the goals of MO. My question is about a certain type of argument in algebraic geometry which exploits the abundance of dense sets ...
9
votes
2answers
345 views

Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
1
vote
1answer
415 views

Standard conjectures on positive characteristic

In this MO answer of M. Bondarko, he says: "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..." and in Remarks on Grothendieck's ...
2
votes
1answer
124 views

Nearby cycles and specialisation - properties

I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
1
vote
2answers
151 views

Plucker embedding and tautological/universal quotient bundle

Let $G$ be a Grassmannian and $Q$ the tautological/universal quotient bundle of $G$. As far as I understand, the associated tautological quotient line bundle for the Plucker embedding of the ...
3
votes
2answers
165 views

Pseudo-automorphisms on Fano varieties

Is every pseudo-automorphism (self-birational map which does not contract any hypersurface) of a smooth Fano variety of Picard rank $1$ equal to a biregular automorphism? Remark: For $\mathbb{P}^n$, ...
1
vote
1answer
535 views

Is Mumford's statement about the representability of some functor wrong?

I am having trouble proving a result in Mumfords book 'Lectures on Curves on an Algebraic surface. It is a statement about the representability of some functor. It is stated on page 108 and says the ...