# 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 $D(f):=\{(đ„_1,âŠ,đ„_đ)ââ€^đ_đ:đ(đ„_1,âŠ,đ„_đ)â 0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
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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 ...
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### 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 $w$, ...
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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 ...
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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 $\text{cl}(K)$: $$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$ ...
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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 ...
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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 ...
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### 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 ...
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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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $T$...
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### 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", ...
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### 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 ...
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### 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 Matsumuraâs ...
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