# Questions tagged [ag.algebraic-geometry]

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

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### Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

This is a cross-post! For the original post on SE (9 upvotes, no answer) see: https://math.stackexchange.com/questions/4475853/is-a-complex-algebraic-set-with-a-zariski-dense-subset-of-algebraic-...
1 vote
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### Intermediate extensions of pure perverse sheaves (BBD 5.4.3)

I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, ...
1 vote
63 views

### Real analytic map with connected fibers

Let $X,Y$ be compact real analytic varieties. Suppose $Y$ is connected and there is a surjective analytic map $f:X\to Y$ such that each fiber of $f$ is connected. How to prove that $X$ is connected as ...
209 views

### Normal schemes and Serre's criterion

Serre's criterion says that for a scheme to be normal is equivalent to it being $R_1$ (i.e. regular in codimension $1$) and $S_2$ (i.e. regular functions on $X-Y$ extend to $Y$ if $Y$ has codimension ...
77 views

### Irreducibility of plane algebraic curves

Given a plane algebraic curve $$y^n + a_1(x)y^{n-1} + \dots +a_{n-1}(x) + a_n(x)y = 0,$$ with a branch point $P_0=(0, y_0)$ of order $n$. Can we prove that this curve is irreducible? What if the ...
204 views

1 vote
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### Computer algebra tools for finding real dimension of an algebraic variety

I have a system of polynomial equations with the unknowns being real numbers. The set of solutions is infinite. What software can I use to compute the real dimension of the solution set? The CAD-based ...
1 vote
106 views

128 views

### Surjective sheaf homomorphisms induced by morphisms of schemes

Let $S$ be a scheme and $X\to Y$ be a morphism over $S$. Then we have an induced homomorphism of sheaves $h_X=\operatorname{Hom}_S({-}, X)\to h_Y=\operatorname{Hom}_S({-}, Y)$ over the small étale ...
150 views

### How to simplify this homotopy totalization coming from an arc-cover into a pullback?

My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know ...
113 views

### Let $V$ be a variety. A point $P \in V$ is nonsingular iff $\dim_k(M_P/M_{P}^{2})=\dim(V)$ [closed]

First of all, we consider $k$ to be an algebraically closed field, and by $M_P$ I denote the maximal ideal of the coordinate ring $k[V]$ at $P$. As for the statement, I have managed to understand how ...
1 vote
101 views

### Abelian subvarieties corresponding to vector subspaces

Let $S$ be a connected smooth projective surface. Let $C$ a smooth curve on $S$ In page 9 of the paper "https://arxiv.org/abs/1704.04187v1" a read the following: Let \begin{equation*} r: ...
1 vote
160 views

### Purity for proper varieties

Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
144 views

### Where can I find that Weil suggested a cohomology theory for characteristic $p>0$?

I have seen that in Grothendieck's paper "THE COHOMOLOGY THEORY OF ALGEBRAIC VARIETIES", he says "The need of a theory of cohomology for 'abstract' algebraic varieties was first ...
149 views

### Compact generation of Voevodsky motives

Let $DM(k,{\mathbb{Q}})$ be the derived category of Voevodsky motives over a field $k$ with $\mathbb{Q}$-coefficients. As a triangulated category is $DM(k,{\mathbb{Q}})$ known or expected to be ...
130 views

### Does the space of hyperplanes in the Grassmannian have a name?

A way of defining the Grassmannian $Gr(k,n)$ is to consider the space of $k\times n$ matrices mod $GL(k)$ transformations on the rows. I'm interested in the space of $k\times 2n$ matrices mod $GL(k)$ ...
138 views

### A group in a neighbourhood of a Zariski dense subgroup

By Borel's theorem, lattices in simple Lie groups are Zariski dense. I expect that a small (in metric sense) deformation of a lattice in a Lie group is also Zariski dense. Suppose we have a Zariski ...
76 views

1 vote
82 views

### Free closed group action on varieties

Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that (1) The $G$ action on $X$ is free and ...
59 views

### Smoothness of homomorphisms between graded algebras

Let $A$ be a finitely generated $\mathbb{C}$-algebra. Let $S^{\bullet}=\bigoplus_{i\geq 0} S^i$ and $T^{\bullet}=\bigoplus_{i\geq 0} T^i$ be two graded $A$-algebras such that $S^0=T^0=A$ and $S^1, T^1$...
1 vote
63 views

Let $V$ be a $2n$-dimensional vector space endowed with a nondegenerate skew-symmetric form $q:V \to V^\vee$. We define the isotropic Grassmannian to be $$X:=G_q(k,V)=\left\{[W] \in \mathbb P \left( \... 2 votes 2 answers 191 views ### Rational solutions to P(x,y)=0 for P reducible over {\mathbb C} There are facts in Mathematics that are so "obvious" and "well-known" that no-one includes a proper proof. An example is: Theorem: If polynomial P(x,y) with rational coefficients ... 2 votes 0 answers 84 views ### Constructions of motivic complex that is only supported on positive degrees It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about ... 3 votes 1 answer 109 views ### Bounded torsion of quotients of affine formal models \DeclareMathOperator\Sp{Sp}Let X=\Sp(A) be a connected smooth affinoid rigid space over a discretely valued non-archimedean field K. Let \mathcal{R} be a valuation ring of K, and fix a ... 4 votes 1 answer 352 views ### Pairing of cotangent and tangent bundles I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie. In math line (2.1.12), Illusie considers the pairing \langle-,-\rangle:\Omega_{X/S}^1\... 0 votes 1 answer 187 views ### Approaching the Riemann-Roch Theorem for algebraic curves I am using "Algebraic Curves: An Introduction to Algebraic Geometry" by William Fulton as a guidline for approaching the Riemann-Roch Theorem for algebraic curves. I have two questions: ... 1 vote 0 answers 106 views ### Cycle class/cohomology class of subvarieties in flat families Let X be a projective variety over \mathbb C and T an irreducible projective \mathbb C-scheme. Let a,b be closed points of T. Suppose we have a flat family Z\to X\times T\to T such that ... 2 votes 1 answer 67 views ### Is the polar dual of a semi-algebraic convex body also semi-algebraic? Call a convex body C\subset\Bbb R^n semi-algebraic if it can be written as$$(*)\quad C=\bigcap_{i\in I}\, \{x\in \Bbb R^n\mid p_i(x)\le 0\}$$with polynomials p_i\in\Bbb R[X_1,...,X_n] and a ... 1 vote 0 answers 129 views ### Complex quadric as a symmetric space It is known that a smooth complex quadric is a symmetric space. For example, it is$$\operatorname{Spin}(n+2)/G$$where G is the maximal parabolic subgroup. I want a reference for more details and ... 1 vote 1 answer 149 views ### Difference between K(1)-local K theory and l-adic completion of etale K theory Let X be an scheme. Fix a prime l which is invertible in X. Consider the K(1)-localization at prime l of algebraic K theory L_1K(X) and l-adic completion of etale K theory K^{et}(X). ... 2 votes 1 answer 135 views ### Is the restriction of the Cartan 3-form on conjugacy classes exact? Let G be a complex semisimple group and \mathcal{O} \subset G a conjugacy class, i.e. \mathcal{O} = \{gag^{-1} : g \in G\} for some a \in G. Let \Omega be the Cartan 3-form on G defined by ... 2 votes 0 answers 104 views ### Tannakian recovery of a group from other tensor abelian categories Classical Tannakian reconstruction recovers an affine group scheme G over k from the category of its linear representations over a field k (as the automorphism group of the forgetful functor to ... 3 votes 1 answer 181 views ### When does a holomorphic symplectic manifold compactify to a Poisson manifold? Let X be a complex manifold endowed with a holomorphic closed 2-form \omega whose associated map \omega : TX \to T^*X is invertible. Can we always embed X as an open subset of a compact ... 0 votes 0 answers 115 views ### Reference Request: sheves of abelian groups over a smooth projective variety Can someone point some good reference (books or lecture notes) for these topics: Let X a smooth projective variety over an algebraically closed field Sheaves of abelian groups over X Quasi-coherent ... -1 votes 0 answers 109 views ### Projectivization of normal bundle We work over the field of complex numbers. Let Y \subset \mathbb P^N be a smooth projective variety. Consider the set I_Y^0 \subset \mathbb P^N \times (\mathbb P^N)^\vee of pairs (y,H) such that ... 3 votes 0 answers 140 views ### Boundedness indices in Voevodsky's smash nilpotence conjecture in family Let X be a smooth projective variety over an algebraically closed field k. Voevodsky introduced the following notion : an algebraic cycle Z in X is smash nilpotent if there exist N>0 such ... 2 votes 1 answer 390 views ### What is the dual of the stable infinity category of perfect complex on smooth proper variety? Fix a commutative ring R. Lurie proved that smooth proper R-linear stable infinity categories are dualizable in \text{Cat}^\text{perf}_{R,\infty}. For a smooth proper variety X over R, what ... 1 vote 1 answer 236 views ### What's a right parameter space of abelian varieties over a non algebraically closed fields? Let k be a field of characteristic not 2 or 3. Then the set of elliptic curves over k can be parametrized by the affine variety S=D(4a^3+27b^2)\subset\mathbb{A}^2_k via the family E\to S where ... 0 votes 0 answers 130 views ### Projection map of the Hirzebruch surface Consider the Hirzebruch surface \mathbb{F}_0:=\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1})\xrightarrow{\pi}\mathbb{P}^1. We know that$$\pi_*\mathcal{O}_{\mathbb{F}_0}(n)\...
1 vote
Let $X$ be a tame DM stack over $\mathbb{C}.$ Let $IX$ denote the inertia stack of $X.$ Let $K(IX)$ denote the Grothendieck group of vector bundles on $IX.$ By the discussion on page 20 of Toen's ...
Let $k$ be a field. Let $X$ be a connected tame DM stack over $k.$ Let $IX$ be the inertia stack of $X.$ Then $IX$ is a disjoint union of connected components. Is this always a finite union? If not, ...