Questions tagged [ag.algebraic-geometry]

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

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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 $D(f):=\{(đ‘„_1,
,đ‘„_𝑛)∈℀^𝑛_𝑝:𝑓(đ‘„_1,
,đ‘„_𝑛)≠0\}\subset \mathbb Z_p^n$. 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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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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227 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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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
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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 answers
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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
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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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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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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
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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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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
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275 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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"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
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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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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
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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
225 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
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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
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180 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
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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
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140 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
95 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
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1 answer
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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
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9 votes
1 answer
532 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
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4 votes
0 answers
327 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
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0 answers
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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
197 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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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
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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
115 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
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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
  • 2,673
1 vote
0 answers
117 views

Is every curve on a projective three-fold a homology-theoretic complete intersection of sorts?

Let $C$ be a curve on a smooth projective three-fold $M$ equipped with the restriction of the Fubini-Study metric $\omega$. I'd like to know if there exists a surface $S$ such that for every closed $(...
Vamsi's user avatar
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3 votes
0 answers
171 views

Wondering if Monsky-Washnitzer ever published a result claimed to be forthcoming in a later paper

At the very end of the paper Formal Cohomology I by Monsky and Washnitzer, they write the following: "In some sense, the operator $\psi$ applied to a power series gives it "better growth ...
Vik78's user avatar
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5 votes
0 answers
519 views

Theorem 7.11 in Scholze's $p$-adic Hodge Theory

I was trying to understand the statement and proof of Theorem 7.11 in Scholze's paper "$p$-adic Hodge Theory for Rigid-Analytic Varieties". I'll reproduce part of the statement below: Let $...
Kush Singhal's user avatar
2 votes
1 answer
267 views

Hypercover and hyper descent

I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
Hello's user avatar
  • 23
7 votes
1 answer
252 views

How to construct such a real algebraic curve

Suppose $L$ is a real line in $\mathbb{RP}^2$, my question is: for a given postive integer $n$, is it possible to find a real projective algebraic curve $C$ of degree $n$ with maximum connected ...
Super Sanae's user avatar
1 vote
0 answers
164 views

Moduli stack of l-adic sheaves?

Let us work over a field $k$. Then for any smooth affine group scheme $G$ over $k$, we can consider the stack quotient $BG := [\text{pt} / G]$ which classifies Ă©tale $G$-torsors. Let $\ell$ be a prime ...
user577413's user avatar
8 votes
0 answers
481 views

An algebraic version of the implicit function theorem for integers

$ \def \x {\boldsymbol x} \def \a {\boldsymbol a} \def \Z {\mathbb Z} $ The famous version of the implicit function theorem (IFT) starts with the assumption of continuous differentiability on the ...
Mohsen Shahriari's user avatar
2 votes
0 answers
72 views

Can Coulomb branches have symplectic resolutions?

My question is about Coulomb branches of a $3D$ $\mathcal{N}=4$ supersymmetric gauge theory, in the sense of Bravermann, Finkelberg and Nakajima Towards a mathematical definition of Coulomb branches ...
jg1896's user avatar
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4 votes
1 answer
582 views

Coherent sheaves, Serre’s theorem and ext groups

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$. Then, the ...
Walterfield's user avatar
0 votes
0 answers
115 views

Roots in indefinite lattice of K3 surfaces

Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$). Inside we have ...
user135743's user avatar
1 vote
1 answer
124 views

Hyperbolicity and inequality for variety of general type

$\DeclareMathOperator\BMY{BMY}$Let $(X,H)$ be a smooth, complex, projective, polarized variety of dimension $n\geq3$, whose canonical bundle $K_X$ is big and nef. Is it know whether the inequality $\...
Armando j18eos's user avatar
1 vote
0 answers
83 views

Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology

The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...
Schemer1's user avatar
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1 vote
0 answers
134 views

Definition of nearby cycle over an affine line

In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
Allen Lee's user avatar
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1 vote
0 answers
117 views

Is there any sufficient or equivalent condition for the invertibility of a regular map, i.e. a self map of $\mathbb{R}^m$ with polynomial components?

Let $P:\mathbb{R}^m\to \mathbb{R}^m$ be a regular map, i.e. a map whose components are polynomials. I was wondering whether we can say anything about the the component polynomials, their degrees or ...
Learning math's user avatar
0 votes
1 answer
158 views

Does going-down theorem hold for local homomorphism of finite flat dimension?

Let $f:(A,m)\rightarrow (B,n)$ be a local homomorphism of Noetherian local rings of finite flat dimension. Does the going-down theorem hold for $f$? If yes, then by Theorem 15.1 in Matsumura’s ...
Boris's user avatar
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