# Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

16,277
questions

**3**

votes

**1**answer

60 views

### Surjective étale morphisms étale locally split

The actual question is slightly more general than that in the title:
Let $p: U\to Y$ be a surjective étale morphism and $Y\to X$ be a finite morphism of schemes. Is there an étale cover $V\to X$ (...

**3**

votes

**0**answers

75 views

### Bialynicki-Birula decomposition for singular projective variety

Let us have a (possibly singular) irreducible projective variety $X$ over $\mathbb{C}$, with an algebraic $\mathbb{C}^*$-action that has finitely many fixed points $\{x_1,\dots,x_n\}$. One can define ...

**2**

votes

**0**answers

95 views

### Family of elliptic curves in $\mathbb P^3$

For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$
in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$.
Let
$$C=l_{12} \cup ...

**0**

votes

**0**answers

46 views

### Two components on the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$

I have two questions adapted from Exercise 24.3.1 in the book Mirror Symmetry.
Consider the moduli of stable maps $\overline{\mathcal{M}}_{2,0}(\mathbb{P}^1,3)$, where a typical element is a degree $...

**0**

votes

**1**answer

47 views

### Slicker computation of the Lie algebra of the symplectic group (and computing differentials of matrix equations of polynomials)

Let $\mathbb{k}$ be an algebraically closed field. The symplectic algebraic group is given by
$$
\text{Sp}(2n,\mathbb{k})=\{M\in\text{Mat}_{2n}(\mathbb{k})\mid J=M^TJ M\}\quad\text{where}\quad J=\...

**5**

votes

**0**answers

168 views

### An explicit description of $X(3)$ and its universal generalized elliptic curve

I'm struggling with the proof of 2.21 of Saito's "Fermat's Last Theorem".
Let $\omega$ be a primitive 3rd root of unity, $X(3) = \mathbb{P}^1_{\mathbb{Q}(\omega)}$, and $E = \{ X^3 + Y^3 + Z^3 - 3 \...

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votes

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181 views

### When are the cotangent and tangent sheaves isomorphic?

Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...

**3**

votes

**0**answers

41 views

### Restriction of semistable sheaf to hyperplane of cover

Let $X$ be a smooth projective variety of dimension at least $2$ over $\mathbb{C}$. Let $\mathcal{O}_X(1)$ be a very ample line bundle and $E$ be a $\mu$-semistable sheaf on $X$. Then a theorem of ...

**9**

votes

**0**answers

227 views

### Grothendieck categories and their morphisms

I am not an algebraic geometer in the first place, and I am mainly familiar with topology and category theory. Recently I am studying Grothendieck categories and I am struggling with getting ...

**6**

votes

**0**answers

137 views

### Explicit construction of the Jacobian of a curve

Let $k$ be an algebraically closed field (of arbitrary characteristic), and $C$ a smooth projective curve over $k$, given by defining equations in projective space. I am looking for an algorithmic ...

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votes

**0**answers

71 views

### Understanding the Hasse-Weil Bound [closed]

While studying Number Theory, I recently encountered with the Hasse-Weil bound,but I know almost nothing about algebraic geometry, and so I don't think I fully understood it.
Does this statement have ...

**3**

votes

**0**answers

89 views

### Where general mixed Galois representations are defined?

I am interested in etale cohomology of varieties, and respectively, in mixed $\mathbb Q_{\ell}$-adic Galois representations over finitely generated fields. What is the canonical reference for this ...

**8**

votes

**0**answers

105 views

### Approximating zero sets of real polynomials with “less complicated” polynomials

Let $K \subset \mathbb{R}^n$ be a compact subset, and let $P(x_1,\dots,x_n)$ be a real multivariable polynomial of degree $d$, whose vanishing set we denote by $Z_P$. Is it plausible to approximate $...

**14**

votes

**2**answers

807 views

### What is the precise relationship between pyknoticity and cohesiveness?

Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement ...

**2**

votes

**1**answer

158 views

### Help with $\mathbf{Q}_{\ell}$ sheaves

Let $X\to S$ be a morphism of smooth connected varieties over an algebraically closed field $k$; let $j:\eta\to S$ be the inclusion of the generic point into $S$ (not a geometric generic point) and ...

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votes

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62 views

### 37.5 is a 20% markup from which number? Explain how you found that number? [closed]

37.5 is a 20% markup from which number? Explain how you found that number?

**7**

votes

**0**answers

105 views

### when the representation space is smooth manifold

In the context of Goldman's paper The symplectic nature of fundamental groups of surfaces:
Consider a closed oriented surface $S$ with fundamental group $\pi$, and let $G$ be a connected Lie group. ...

**2**

votes

**0**answers

53 views

### $m$-regularity of sheaves

This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...

**-1**

votes

**1**answer

94 views

### Connections on vector bundles over elliptic curves - concrete computations

This is linked to my question on math.Stackexchange for which I had no answer.
I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me ...

**3**

votes

**0**answers

111 views

### Is cup product of cycle classes on Noetherian regular excellent scheme compatible with intersection

Let $\mathcal{X}$ be a Noetherian regular integral excellent scheme. Let $Y$ and $Z$ be algebraic cycles of codimension $c$ and $d$ on $\mathcal{X}$.
Let $n$ be a positive integer invertible on $\...

**2**

votes

**3**answers

249 views

### Algebra of regular functions on the quadratic cone and SU(2) representations

I was reading the paper "Short Star-Products for Filtered Quantizations" by Pavel Etingof, were in the introduction he claims that the algebra of regular functions on the quadratic cone $X$ in $\...

**4**

votes

**1**answer

130 views

### Which $p$-adic valuations of Weil numbers (that is, eigenvalues of Frobenius) are possible?

Let $C$ be a smooth projective curve over a finite field $\mathbb F_q$, $q$ is a power of the characteristic $p$. It is well-known that if $\alpha$ is an eigenvalue of Frobenius acting on $H^1_{et}(C,\...

**2**

votes

**1**answer

243 views

### Ideal corresponding to the inverse image of a submanifold

I am currently learning about algebraic viewpoint on closed embedded sub(super)manifold. In particular, I am struggling with something which should be ''easy to see''. Namely, I refer to the lemma ...

**1**

vote

**1**answer

107 views

### Are morphisms from affine schemes to Artin stacks affine morphisms?

It is explained in this MO discussion that if one has a morphism $f:X\rightarrow Y$ of schemes such that $X$ is an affine scheme, then $f$ need not be an affine morphism. However, if $Y$ is separated, ...

**10**

votes

**1**answer

234 views

### Real manifolds and affine schemes

I noticed the following strange (to me) fact. If $M$ is a real manifold (smooth or not) and $R = C(X, \mathbb{R})$ is the ring of real functions (smooth functions in the smooth case) then the affine ...

**0**

votes

**0**answers

54 views

### Existence of integral open subscheme with some nice properties

Let $R$ be a discrete valuation ring and $Spec(R):=S= \{\sigma, \eta\}$ it's affine scheme with closed point $\sigma$ and generic $\eta$. Let $f:Y \to S$ a dominant morphism of schemes of finite type. ...

**1**

vote

**1**answer

263 views

### Why are Serre functors always exact?

Let $k$ be a field and $\mathcal{T}$ be a $k$-linear triangulated category with finite dimensional spaces of morphisms. Bondal and Kapranov proved that every Serre functor on $\mathcal{T}$ is exact (...

**0**

votes

**0**answers

50 views

### Good reference on measure theory on Borel algebras of schemes

I am looking for a good reference on measure theory on Borel algebras of schemes, preferrably one that's available online since every library is in lockdown right now.
Any suggestions?

**5**

votes

**0**answers

247 views

### Geometric interpretation of nonconnective, non-coconnective chain complexes / spectra?

Let's stipulate that
Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about ...

**2**

votes

**1**answer

147 views

### Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...

**2**

votes

**0**answers

112 views

### Smooth subvarieties through a singular point

Suppose that $X$ is a singular variety of dimension $n$, with singular locus of dimension $a$. Does there always exist a smooth subvariety $V$ of given dimension $m$ (where $a < m < n$) which ...

**2**

votes

**1**answer

178 views

### Formal neighbourhood of a closed subscheme

Let $X$ be a variety and $Y \subset X$ a closed subvariety.
Edit: Assume they are both smooth.
Denote $N_{Y / X}$ the normal bundle of $Y$ in $X$. The formal neighbourhood of $Y$ in $X$ is the ...

**3**

votes

**0**answers

95 views

### How to make algebraic dependence explicit

Here is a possibly standard question, from someone who is in no way an expert. The scenario is taken from Shafarevich, Basic Algebraic Geometry 1, ch. 1, §6.3, proof of Thm. 1.25(ii).
Let us have a ...

**0**

votes

**0**answers

47 views

### The rotation angle of ellipse. Am I making a mistake? [closed]

we have the general formula $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$.
The semi-major and semi-minor axes are $\sqrt{\frac{2(4ACF+BDE-AE^2-CD^2-FB^2)}{(B^2-4AC)(A+C+-\sqrt{(C-A)^2+B^2})}}$ using matrices and other ...

**6**

votes

**1**answer

162 views

### Localization of symmetric monoidal categories and geometry

I have a series of vague questions, related to localization of symmetric monoidal categories.
Here is the context. Say we are working over a field of characteristic zero. Then the "one category ...

**11**

votes

**1**answer

305 views

### De Rham and Koszul complexes

Consider the algebraic de Rham complex of the $n$-dimensional plane: this is merely
$$\ldots\rightarrow Sym(V^*)\otimes\bigwedge^{k}V^*\rightarrow Sym(V^*)\otimes\bigwedge^{k+1}V^*\rightarrow\ldots
$$...

**3**

votes

**1**answer

127 views

### Translates of abelian subvarieties

Suppose $A$ is an abelian variety over an algebraically closed field $k$. I'm confused about the notion of translates of abelian subvarieties. I looked over a few related papers, but I couldn't find a ...

**2**

votes

**0**answers

103 views

### A Lefschetz style formula for the $\ell^\infty$ torsion of an Abelian variety over a finite field

Let $A/\mathbb F_q$ be an abelian variety over a finite field. Define $A_\ell = A[\ell^\infty](\mathbb F_q)$, the $\ell^n$ ($n\geq 0)$ torsion points defined over the base field. I can assume $\ell \...

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votes

**0**answers

112 views

### Is the scheme of morphism dense? [closed]

Let $X$ and $Y$ projectives algebraic varieties with $X \subset Y$ open and dense subset, then $Hom(\mathbb{P}^{1},X) \subset Hom(\mathbb{P}^{1},Y)$ is an open dense subset ? ($\mathbb{P}^{1}$ is a ...

**3**

votes

**2**answers

250 views

### Is any constant Zariski sheaf already a Nisnevich sheaf?

Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (...

**1**

vote

**1**answer

82 views

### isogenies between elliptic curves with multiplicative reduction

Let $ K $ be a $ p $-adic field. Suppose we have an isogeny of elliptic curves $ \phi : E \to E' $ defined over $ K $, where $ E $ and $ E' $ both have multiplicative reduction.
1) Is there anything ...

**1**

vote

**0**answers

81 views

### The normal cone to a singular subvariety

Let $Y$ be a smooth variety, and let $X\subset Y$ be a subvariety. If $X$ is smooth, then $C_XY=N_XY$ is a quotient of $TY|_X$. Is the same still true if $X$ is singular? I.e. is there always a ...

**7**

votes

**1**answer

187 views

### Finding $Q(\sqrt{-2})$-rational points on $X_0(33)$

Let $K = Q(\sqrt{-2})$. How can I compute the $K$-rational points on the modular curve $X_0(33)$?
Recall that $X_0(33)$ is of genus $3$ and has the following affine model,
$$y^2 +(-x^4-x^2-1)y = 2x^...

**1**

vote

**0**answers

115 views

### Transcendance in function fields

Denote by $\Omega$ the completion of an algebraic closure of $\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$ for the valuation $-\deg$. Let $(a_n)_n$ be a sequence of $\overline{\mathbb F_q(T)}\...

**1**

vote

**0**answers

162 views

### Projective embeddings of quotients of normal varieties

Let $X$ be a normal complex projective variety of dimension $m$, $G$ be a finite subgroup of $\mathrm{Aut}(X)$, and $Y = X / G$ be the quotient. I am particularly interested in the case where $X$ is a ...

**3**

votes

**0**answers

145 views

### Is the Cassels “$x - \theta$” map algebraic in some sense?

Setup: Let $k$ be a field of characteristic $0$, let $f(x) \in k[x]$ be a monic separable polynomial of degree $n \geq 4$, and let $\theta$ denote the image of $x$ under the map $k[x] \to K_f := k[x]/(...

**2**

votes

**0**answers

91 views

### Morphism between jet spaces smooth

In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:
Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...

**1**

vote

**1**answer

99 views

### Is $\mathbb{Q}$-factoriality preserved under contraction?

Let ($X$,$\Delta$) be projective klt pair and $f \colon X \rightarrow Z$ be contraction of ($K_X + \Delta$) - negative extremal ray $R$.
If $X$ is $\mathbb{Q}$ -factorial and $\mathrm{dim}Z < \...

**3**

votes

**0**answers

81 views

### Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities.
Assume that there is an isomorphism $f:...

**3**

votes

**1**answer

198 views

### The Zariski topology of a variety is determined by its principal Cartier divisors

Let $X$ be a smooth separated integral variety over an algebraically closed field.
Question:
Is it true that a basis for the Zariski topology is given by the family $$\mathcal{U}=\{X\setminus \...