Questions tagged [ag.algebraic-geometry]

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

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Intersection theory on normal crossing algebraic surfaces

Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...
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4 votes
2 answers
452 views

Are algebraic groups over algebraically closed fields Cohen–Macaulay?

$\DeclareMathOperator\CM{CM}\DeclareMathOperator\Spec{Spec}$Let $k$ be an algebraically closed field and let $G$ be an algebraic group over $k$. My question: is $G$ Cohen–Macaulay? If not, are there ...
0 votes
1 answer
205 views

Hilbert scheme of divisors in smooth projective varieties

Let $X$ be a smooth projective variety and $L$ be a line bundle with $H^0(X,L)\neq 0$. Let $D\in |L|$ and $p(t)$ be the Hilbert polynomial of $D$. Assume that any effective divisor $D'\subset X$ with ...
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0 votes
1 answer
309 views

Motivation behind spectral sequences

It is well known that spectral sequence is very important in algebraic geometry and complex geometry, but its definition seems very unnatural. For example, in Voisin's book Hodge theory and complex ...
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1 vote
1 answer
144 views

Two morphisms possess the same Viehweg's variation

Recall the definition of Viehweg's variation intimated from Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces, E. Viehweg Let $f: V\rightarrow W$ be a fiber space (...
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1 vote
0 answers
60 views

Strategy to distinguish a rank 2 vector bundle from an extension class group

If you think this question is very basic, then I apologies my ignorance at the first stance. Suppose $V$ is a rank 2 vector bundle on a rational ruled surface $\pi:\mathrm{X}=:\mathbb{P}(\cal{O}_{\...
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2 votes
1 answer
100 views

Is the Segre embedding of two real varieties a real variety?

$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve ...
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4 votes
1 answer
294 views

How does the rational curve behave under the group action?

Let $X$ be $\mathbb{P}^{2}$(over $\mathbb{C}$) blowing up at 7 points in general position. Denote the points as $p_{1}, p_{2}, …,p_{7}.$ Denote the blowing up map as $\pi_{1}: X\to \mathbb{P}^{2}$. We ...
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2 votes
1 answer
187 views

Fourier-Mukai functors and autoequivalence groups of $G$-equivariant derived categories

I have a few questions about $G$-equivariant derived categories. For my question, I'm assuming $G$ is cyclic. Also, in my case $G$ does not act on $X$, only on $D^b(X)$. Q1: Orlov's Representability ...
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2 votes
0 answers
79 views

Induced map on cohomology by automorphisms of smooth hypersurfaces

Recently I am considering a strange question. Consider a complex smooth projective hypersurface $X\subset\mathbb{P}^{n+1}$, then an automorphism $f:X\rightarrow X$ induces an isomorphism on Betti ...
10 votes
0 answers
180 views

Symmetric spaces are quandles. Is this important?

For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
4 votes
1 answer
271 views

Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?

The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as \begin{equation} \det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
2 votes
0 answers
156 views

Proof of the projection formula (for cohomology of $\mathbf{P}V$)

Let $V\to X$ be a vector bundle (over say a scheme). Then the cohomology of its projectivisation is $$\text{H}^*(\mathbf{P}V)\ =\ \text{H}^*(X)[t]/(t^{n+1}+c_1(V)t^n+\cdots+c_n(V))$$ as an algebra, ...
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1 vote
0 answers
71 views

Fourier-Mukai kernels for Fano threefolds

Let $Y_1$ and $Y_1'$ be index two degree one Fano threefolds. Suppose we have a Fourier-Mukai equivalence $\Phi_P : \mathrm{D}^b(Y_1) \to \mathrm{D}^b(Y_1')$. Can anything be said about the kernel $P$,...
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3 votes
1 answer
151 views

Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder https://arxiv.org/pdf/2207.06477.pdf Lemma 6.18, $D^b(...
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3 votes
1 answer
194 views

Embeddings of reductive groups over algebraically closed fields

Let $K/k$ be an extension of fields, not necessarily algebraic; let $G$ and $H$ be split, reductive groups over $K$; and let $f : H \to G$ be an embedding of groups. Do there exist split, reductive ...
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3 votes
2 answers
303 views

Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$

Let $X$ be a complex manifold, there is a natural map $f:H^2(X,\mathbb C)\to H^2(X,\mathcal O)$ induced by the inclusion map $\mathbb C\hookrightarrow \mathcal O$ which coincides with the natural ...
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1 vote
0 answers
111 views

Example of a brick-infinite, tame, triangular algebra of global dimension$\geq 3$

I'm trying to compute some examples and I'm unable to come up with a following example: What is(are) the example(s) of an acyclic quiver $Q$ with relations such that the 2-Kronecker quiver is NOT a ...
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2 votes
1 answer
193 views

Rational curves on the image of the pluricanonical maps

Let $X$ be a compact complex manifold with canonical bundle $K_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but not maximal, i.e., $X$ is not of general type). Let $\varphi_m : X \...
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-2 votes
1 answer
178 views

Topologies in the vicinity of Euclidean space

Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$. Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
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3 votes
1 answer
264 views

Variant of Wahba's problem

Wahba's problem is the following: $$\min_R \sum_{k=1}^K \|v_k - Rw_k\|^2$$ where $v_k$ and $w_k$ are arbitrary $3\times 1$ vectors, and $R$ is a rotation matrix (i.e., orthogonal with $\det(R)=1$). A ...
12 votes
2 answers
1k views

What exactly do the standard conjectures in characteristic zero refer to?

As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an ...
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0 votes
0 answers
84 views

Complex dimension of zeros of vanishing ideal vs real dimension

Let $S \subseteq \mathbb{R}^n$ be a subset of real points and $I(S)$ be the vanishing ideal of $S$ in $\mathbb{R}[x_1,\dotsc,x_n]$. Is $\dim V_{\mathbb{R}}(I(S)) = \dim V_{\mathbb{C}}(I(S))$? I.e., is ...
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1 vote
0 answers
81 views

Positive $(p,p)$-current and subvariety

Let $X$ be a complex manifold. Any dimension $p$ subvariety(or reduced analytic subscheme) will determine a real closed positive $(p,p)$-current. But the converse is not true. My question: Is there a ...
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1 vote
0 answers
103 views

Question regarding forgetting morphism of the moduli spaces of pointed rational curves

I am trying to understand the moduli space of pointed rational curves with $n$ marked points, $\overline{M_{0,n}}$. I have following doubts Let $\pi_i:\overline{M_{0,n+1}}\to\overline{M_{0,n}}$ be the ...
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1 vote
0 answers
95 views

Joins of (closed) subschemes and Zariski-local Z-functors

$\newcommand\Aff{\mathrm{Aff}}\newcommand\cRing{\mathrm{cRing}}\newcommand\Sch{\mathrm{Sch}}$Equip $\Aff = \cRing^\text{op}$ with the Zariski Grothendieck-topology. There are nested categories: $$\Aff\...
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1 vote
0 answers
114 views

More on points on a curve of genus 3

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find two ...
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4 votes
1 answer
156 views

About Fulton's Intersection theory Appendix Lemma A 4.1

The assumption for Lemma A.4.1 is $A \to B$ is flat. The second assumption is that $A$ and $B$ are Artinian rings. From this Lemma A.4.1 states that $l_B(B) = l_A(A) \cdot l_B(B/mB)$ where $m$ is the ...
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0 votes
0 answers
104 views

Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off? Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...
5 votes
0 answers
78 views

Does a discriminant condition on $f(x,y)$ imply that $f$ is weighted homogeneous?

[This is an updated version of https://math.stackexchange.com/questions/4522399/.] Let $f = \sum_{i=m}^n f_iy^i \in \mathbb{C}[x,y]$ be a polynomial (where $f_i \in \mathbb{C}[x]$ with $f_m,f_n \ne 0$ ...
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2 votes
1 answer
287 views

Points on curves of genus 3

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find a point ...
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4 votes
0 answers
165 views

Rational solutions to Catalan's equation

Famous Catalan's conjecture, now a theorem proved by Mihăilescu, states that the only solution in the natural numbers of the equation $$ x^{a}-y^{b}=1. $$ for $a, b > 1$ and $x, y > 0$ is $x = 3,...
3 votes
1 answer
174 views

Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

Let $f:X\to S$ be a morphism between algebraic varieties which are smooth over a field of characteristic zero. We define the (derived) direct image functor $f_+:\mathsf{D}^b(\mathcal{D}_X)\to \mathsf{...
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1 vote
0 answers
164 views

Projective scheme over the integers

Let $X$ be a projective scheme over $Spec(\mathbb{Z})$. Let $X_{p}$ be the reduction at $p$ of $X$. If for any prime $p$, $X_{p}$ is normal, can we deduce $X$ is normal? Or any counterexamples?
1 vote
0 answers
90 views

Non vanishing of a cohomology class associated to a nef vector bundle

Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$: $$ \sum_{k=0}^nt^{n-k}\int_Xc_k(...
4 votes
0 answers
125 views

Surface with $\Omega_X$ globally generated and singular Albanese image

This question is inspired by abx's comment to my previous question MO430933. Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$...
0 votes
0 answers
160 views

Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...
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3 votes
1 answer
326 views

Trigonometric Diophantine equation

Is there a general method to solve the equation $P(x_1,x_2,...,x_n)=0$ with $P$ is a polynomial in $n$ variables with integer coefficients and $x_k=\cos(q_k\pi)$ with $q_k$ is a rational number? This ...
1 vote
1 answer
267 views

Realize as homology a given polynomial ring

I am wondering if one can realize a polynomial ring as the homology of some chain complex in the same sense that the homology groups of a space with a cell complex structure is the homology of its ...
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6 votes
1 answer
232 views

D-modules on singular varieties; forgetful functors, and t-structures

Let $Z$ be a singular variety over the complex numbers with a closed embedding $i: Z \hookrightarrow X$ into a smooth variety $X$. One can define the derived category $\mathcal{D}(Z)$ of D-modules on $...
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3 votes
1 answer
154 views

Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection. Can one understand how the compactified Jacobian of $C$ looks like? For example, the ...
2 votes
1 answer
82 views

Library/Database of parametric polynomial systems

Could anyone please recommend a known website where I can find a database/library that has systems of polynomial equations with $n$ variables and $m$ parameters? I need some real examples to test my ...
1 vote
0 answers
118 views

Contracting an effective Cartier divisor to a point

I am trying to understand contractibility of a subspace, in the case of contracting a divisor to a point. More precisely, what I mean by contraction is the following. Given an algebraic space(or ...
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2 votes
0 answers
62 views

How to get a concrete description of $\pi_*\Omega_{X/S}(Z)|_Z$, when $X \supset Z \to S$ is a finite extension of Dedekind schemes?

Let $X/S$ be a proper, smooth relative curve over a Dedekind scheme $S$, for example, $X = \mathbb{P}^1_S \xrightarrow{\pi} S$. Suppose that $Z \to X$ is a horizontal effective Cartier divisor such ...
2 votes
0 answers
65 views

Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
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9 votes
0 answers
191 views

Fundamental group of the complement to a plane curve with unramified normalization

Suppose that $C\subset\mathbb P^2$ is an irreducible projective curve over $\mathbb C$ such that the normalization morphism $\bar C\to C$ is unramified (i.e., the induced morphism $\bar C\to\mathbb P^...
1 vote
1 answer
145 views

Intersection pairing and birational morphisms

Let $f:X\to Y$ be a birational morphism of smooth projective variety. We assume that $f(V)\simeq U$ isomorphism induced by $f$, where $V\subset X$ and $U\subset Y$ are two Zariski open sets. Let $x\in ...
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2 votes
1 answer
136 views

Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion. Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical ...
2 votes
1 answer
148 views

Intersection of translate of divisors on abelian variety

Setup. Let $K$ be an algebraically closed field of characteristic zero, and let $A/K$ be a simple abelian variety of dimension $n$. Let $\{ x_1,x_2,\dots,x_{m^{2n}}\}$ denote the $m$-torsion points of ...
3 votes
0 answers
122 views

Variety whose secant variety is a cubic hypersurface

Is there a characterization of projective varieties $X\subset\mathbb{P}^n$ whose secant variety is a hypersurface of degree $3$? In the case that the secant variety does not have the expected ...
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