Questions tagged [ag.algebraic-geometry]

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

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If $\pi$ is a prime of a UFD $A$, is $\text{Spec }A$ a coproduct of $\text{Spec }A[\pi^{-1}]$ and $\text{Spec }A_{(\pi)}$ over $\text{Spec Frac }A$?

Let $A$ be a UFD (unique factorization domain) with fraction field $K$. Let $\pi\in A$ a prime. Let $A_{(\pi)}$ be the localization at the ideal $\pi$, and let $A[\pi^{-1}]$ be the localization w.r.t. ...
stupid_question_bot's user avatar
3 votes
2 answers
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Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated

Closely related to this question in MSE, but the difference is that we will set $X$ to be scheme and $\mathcal{F}$ to be quasi-coherent. Let $X$ be a locally ringed space. We say an $\mathcal{O}_X$-...
Z Wu's user avatar
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Formula for Pushforward of structure sheaf for branched coverings

I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces. Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $...
user267839's user avatar
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3 votes
1 answer
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Descent theory for higher sheaves

I am trying to understand the descent condition for sheaves from presheaves. Let Presh(S) be the $(\infty,1)$ category of presheaves on an $(\infty,1)$ site S, and Sh(S) be the corresponding category ...
Pinak Banerjee's user avatar
4 votes
0 answers
368 views

Questions about the Chow varieties

In Lecture 21 of Joe Harris's famous textbook "Algebraic geometry: a first course", he introduced the concept of Chow varieties. In Theorem 21.2, he says that the open Chow variety has ...
LittleBear's user avatar
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Excision in "3264 and all that" by Eisenbud-Harris

In Proposition 1.14, page 25 in the book "3264 and all that Intersection Theory in Algebraic Geometry" the authors define a right exact sequence: $$ Z(\mathbb{P}^1 \times X) \rightarrow Z(X) ...
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On the positivity of cotangent bundle of elliptic surfaces

I am working on the complex numbers field $\mathbb{C}$, for simplicity. However you can relax this assumption if you want. Let $X$ be an elliptic surface, id est there is a proper morphism $\pi\colon ...
Armando j18eos's user avatar
2 votes
1 answer
191 views

Construction refuting the existence of nonisotrivial elliptic curve over $\mathbb{G}_m$

I have some troubles to understand the construction in detail presented here by Daniel Litt used to show that there cannot exist an elliptic curve over $\mathbb{G}_m/k$, $k$ of characteristic $p >3$...
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Diagonal morphism of henselization is an open immersion?

Let $(R,\mathfrak{m})$ be a local ring, denote by $R \rightarrow R^h$ its henselization. Write $S = \operatorname{Spec} R$ and $S^h = \operatorname{Spec} R^h$. Is it true that the diagonal morphism $\...
Hugo Zock's user avatar
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Effective Torelli theorem for K3 surfaces

The proof of the Torelli theorem I've seen goes something like: Put $M$ the moduli space of marked $K3$ surfaces, and $D$ the period domain s.t there is a natural map $$P: M \to D$$. Up to lies, here ...
user135743's user avatar
6 votes
1 answer
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Functor of points of the cone over the Grassmanian

$\newcommand{\Gr}{\operatorname{Gr}}$Let $\Gr(d,n)$ be the Grassmanian (I am assuming the field is $\mathbb{C}$). Let us denote by $\widehat{\Gr(d,n)}$ the cone of the Grassmanian under the plucker ...
Charvaka's user avatar
5 votes
2 answers
226 views

Characterize the space of all ramification divisors of degree $d$

Let $X$ be a compact Riemann surface of genus $g>0$, and let $f\colon X \to \mathbb{P}^1$ be a branched covering of degree $d$. Define the ramification divisor $R_f$ on $X$ by $f$, where $\deg R_f =...
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Action on Enriques surface by sections of Jacobian fibration

A question about a statement in Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups: The setup: Let $\pi: Y \to \mathbb{P}^1$ be a special elliptic pencil of complex Enriques ...
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Tate–Shafarevich group and $\sigma \phi(C)=-\phi \sigma(C)$ for all $C \in \operatorname{Sha}(E/L)$

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $\Gal(L/K)$. Let $E/K$ be an elliptic curve defined ...
Duality's user avatar
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3 votes
1 answer
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Hyperelliptic integrals

I am learning about hyperelliptic curves and hyperelliptic integrals. I encountered some problems when reading the book by Gesztesy and Holden (F. Gesztesy, H. Holden, Soliton Equations and Their ...
mxjia's user avatar
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1 answer
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How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?

Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
Richard's user avatar
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Cohomology ring of $\mathbb{P}(\mathcal{O}(-1)\oplus \mathcal{O})$

Let $\mathcal{O}(-1)$ be the Hopf bundle over $\mathbb{C}\mathbb{P}^\infty$. Let $\mathcal{O}$ be the trivial rank one bundle. Consider the projectivization of the rank two bundle $\mathcal{O}(-1)\...
asv's user avatar
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elliptic curves on general 3-folds of degree 7

Do there exist elliptic curves on a general 3-fold hypersurface $X_7 \subset \mathbb{P}^4$ of degree $7$? Clemens proved that for $d \ge 8$ there are no elliptic curves on the general hypersurface $...
Ben C's user avatar
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7 votes
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What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...
David Corwin's user avatar
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223 views

About a formula in Lawrence-Venkatesh's proof on Mordell conjecture

In Lawrence-Venkatesh, the lemma 2.10 states that For number fields $L/K$, and a representation $\rho:G_L\to GL_n(\mathbb{Q}_p)$ that is crystalline at all primes above $p$ and pure of weight $w$, ...
Phanpu's user avatar
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216 views

Infinity stacks

I was going through some notions of stacks and higher stacks on nLab. $\infty$-stacks are usually $(\infty,1)$-sheaves which take values in $\infty$-groupoids. Now to recall, $(\infty,1)$-sheaf is a ...
Pinak Banerjee's user avatar
4 votes
1 answer
396 views

Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?

Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$: $$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$ ...
Snacc's user avatar
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2 votes
1 answer
249 views

Uniqueness of sum of squares representation

Given a polynomial $f(x) \in \mathbb{R}[x] = \mathbb{R}[x_{1},\dots,x_{n}]$. We say $f(x)$ is sum of squares(SOS) if there are polynomials, $p_{1},\dots,p_{k}$ such that $f = p_{1}^{2} + \dots+p_{k}^{...
wsz_fantasy's user avatar
5 votes
1 answer
228 views

Local triviality of torsors for relative reductive groups

Let $X \to S$ be a relative (smooth proper) curve, and $G \to X$ a reductive group scheme. The following two results are well-known: (Drinfeld-Simpson) For arbitrary $S$, if $G$ is defined over $S$, ...
C.D.'s user avatar
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Do amoebas obtain extra tentacles as we take the tropical limit?

Original Question In this question, we'll restrict ourselves to plane curves. Define the $t$-amoeba of a polynomial $p(z,w) = \sum_{i,j \in \mathbb{N}} a_{ij} z^i w^j$ to be the set $\mathcal{A}_t(p) =...
mijucik's user avatar
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1 answer
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Is this toric variety always smooth?

Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray in $\sigma$. Let $U_{\rho}$ be defined as $\operatorname{Spec}(k[\...
Boris's user avatar
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1 answer
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Very ample + effective = ample?

Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) ...
Calculus101's user avatar
2 votes
0 answers
139 views

Inclusion of boundary strata of moduli of curves: induced map on tangent spaces

$\DeclareMathOperator\Ext{Ext}$Let $C \in \bar{\mathcal{M}}_g$ be a nodal curve. It is a classical result that the tangent space of $\bar{\mathcal{M}}_g$ at $C$ is given by \begin{align*} T_C \bar{\...
Matthias's user avatar
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6 votes
0 answers
491 views

Genus of a number field

I'm reading Algebraic Number Theory by Neukirch. In chapter 3, he defines the genus of a number field as $$ g = \log \frac{ |\mu (K)| \sqrt{|d_K|}}{2^{r} (2\pi)^{s}} $$ where $|\mu(K)|$ is its ...
Leonardo Lanciano's user avatar
9 votes
0 answers
273 views

Sign of the Euler characteristic of a variety of general type

Let $X$ be a smooth projective complex variety of general type, minimal ($=K_X$ nef), of dimension $n$, and let $e(X)$ be its (topological) Euler characteristic. If $n=1$ or $2$, we have $(-1)^ne(X)&...
abx's user avatar
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0 votes
0 answers
67 views

"Approximating" ring of semi-invariants

I'm trying to calculate the semi-invariant ring for certain types of quivers. For a very brief introduction to semi-invariant rings of quiver please have a look at this wikipedia article at the ...
It'sMe's user avatar
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7 votes
1 answer
488 views

Is there a direct translation between Tropical and Algebraic geometry?

I am thinking of doing research in Tropical Geometry for my senior thesis, and I was wondering to what degree we can translate problems in Algebraic Geometry to Tropical Geometry. We know that there ...
mijucik's user avatar
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0 answers
149 views

Is this closed subscheme a toric variety?

Let $k$ be an algebraically closed field. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$ and let $\rho$ be a ray of $\sigma$. Say $\rho=\sigma\cap H_m$, where $H_m$ is the plane in $...
Boris's user avatar
  • 491
4 votes
0 answers
204 views

Economic equilibrium and tropical geometry

There is a famous saying in economics: When everyone pursues his or her own interests, there is an invisible hand that brings the market to equilibrium. However, this is not always the case. Here is ...
Surpass2019's user avatar
3 votes
0 answers
224 views

Algebraic Fukaya categories and mirror symmetry

Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
Robert Hanson's user avatar
9 votes
0 answers
300 views

Coordinate ring of universal centralizer (BFM space)

In the paper titled Equivariant (K-)homology of affine Grassmannian and Toda lattice, the authors, Roman Bezrukavnikov, Michael Finkelberg, and Ivan Mirković, derived the coordinate ring of each ...
Yunsong WEI's user avatar
2 votes
0 answers
179 views

Proposition 4.3.8 Qing Liu about flat morphisms of schemes

I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves"). The statement is: Let $Y$ be a scheme having only a finite ...
BernyPiffaro's user avatar
0 votes
0 answers
78 views

Restrictions of a morphism that is piecewise smooth

My lecture notes of classical algebraic geometry on complex field has presented a following result. Theorem. Let $X$ and $Y$ be (quasi-projective irreducible) varieties, and $f \colon X \to Y$ a ...
Lasting Howling's user avatar
1 vote
0 answers
137 views

What is the interpretation of the reduction modulo $p$ of the modular curve $X(N)$ for $p$ dividing $N$?

Let $N>3$ be an integer. The modular curve $X(N)$ is the compactification of the scheme parametrising triples $(E,t,t)$ where $E$ is an elliptic curve defined over a field of characteristic 0, and $...
Xavier Roulleau's user avatar
3 votes
0 answers
94 views

A question on the averages of Kloosterman sums

Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is, For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound $$\sum_{...
hofnumber's user avatar
  • 553
1 vote
1 answer
168 views

Is the map on tame fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

$\DeclareMathOperator\Spec{Spec} $Let $k \subset L$ be two algebraically closed fields of characteristic $p$. Let $U \subset \mathbb P^1_k$ be a smooth quasi-projective curve and let $U_L$ denote the ...
Cloud63's user avatar
  • 13
9 votes
1 answer
531 views

Irrationality of cubic threefold (before Clemens and Griffiths)

I came across this notice, which seems to say Fano proved that a general cubic threefold is irrational back in 1940s. I'm interested in seeing this work, especially a proof without intermediate ...
AG learner's user avatar
  • 1,356
4 votes
0 answers
324 views

Non-Noetherian (classical) algebraic geometry

My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
Daniel W.'s user avatar
  • 365
2 votes
0 answers
91 views

Pullback of canonical bundle along a group quotient

Suppose $X$ is a smooth variety over $\mathbb C$ with a free action of an affine algebraic group $H$ over $\mathbb C$. Then there is the quotient map $$ p: X \to Y := X/H. $$ Suppose $Y$ is smooth ...
Qixian Zhao's user avatar
4 votes
1 answer
166 views

Are the two notions of free $\mathbb{G}_a$-actions equivalent?

Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation $$\...
Display Name's user avatar
0 votes
1 answer
194 views

Stable curve local complete intersection

Let $C$ be a stable curve over base field $k$. How to show that $C$ is local complete intersection purely algebraically? I'm emphasizing pure algebraically here because the only proof of this ...
user267839's user avatar
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0 votes
0 answers
150 views

Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator

In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
Schemer1's user avatar
  • 769
1 vote
1 answer
123 views

inverse image of a family of sheaves

Les $X$, $Y$, $S$ be noetherian schemes, $f:Y\to X$ a morphism and $\mathcal{F}$ a coherent sheaf on $X\times S$, flat on $S$. Is it true that $(f\times I_S)^*\mathcal{F}$ is flat on S for every ...
Hephaistos's user avatar
1 vote
1 answer
113 views

An etale cover of a semiperfect ring

Assume that $R$ is a semiperfect ring in characteristic $p$, i.e the frobenius is surjective on $R$. I think one can prove that an etale cover of $R$ should again be semiperfect by considering the ...
ALi1373's user avatar
  • 65
4 votes
0 answers
119 views

Coulomb branches which are not of cotangent type

To each $3d \, N=4$ supersymmetric quantum field theory $\mathcal{T}$, there is a related space called the Coulomb branch of this theory, $\mathcal{M}_C(\mathcal{T})$ (it is a piece of the moduli ...
jg1896's user avatar
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