# Questions tagged [ag.algebraic-geometry]

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

21,434
questions

4
votes

1
answer

157
views

### 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. ...

3
votes

2
answers

325
views

### 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 $\mathcal{O}_X$-...

1
vote

0
answers

120
views

### 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 $...

3
votes

1
answer

282
views

### 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 ...

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 ...

0
votes

0
answers

214
views

### 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) ...

1
vote

0
answers

170
views

### 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 ...

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 $p >3$...

3
votes

1
answer

307
views

### 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 $\...

1
vote

0
answers

87
views

### 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 ...

6
votes

1
answer

254
views

### 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 ...

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 =...

1
vote

0
answers

184
views

### 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 ...

0
votes

1
answer

274
views

### 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 ...

3
votes

1
answer

209
views

### 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 ...

2
votes

1
answer

144
views

### 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 $D(f)$...

0
votes

0
answers

120
views

### 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)\...

2
votes

0
answers

99
views

### 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 $...

7
votes

0
answers

240
views

### 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 ...

5
votes

0
answers

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 $w$, ...

0
votes

0
answers

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 ...

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 $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...

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}^{...

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 $S$, ...

7
votes

1
answer

491
views

### 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) =...

1
vote

1
answer

232
views

### 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[\...

0
votes

1
answer

227
views

### 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) ...

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{\...

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 ...

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)&...

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 ...

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 ...

1
vote

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 $...

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 ...

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 ...

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 ...

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 ...

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 ...

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 $...

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_{...

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 ...

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 ...

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 ...

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 ...

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
$$\...

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 ...

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, ...

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 ...

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 ...

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 ...