# Questions tagged [ag.algebraic-geometry]

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

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

20,024
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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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$\DeclareMathOperator\CM{CM}\DeclareMathOperator\Spec{Spec}$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 ...

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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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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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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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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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$\newcommand{\complex}{\mathbb{C}}\newcommand{\real}{\mathbb{R}}\newcommand{\proj}{\mathbb{P}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Seg{Seg}$I apologize in advance for my naïve ...

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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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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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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 ...

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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: ...

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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)^{⊗...

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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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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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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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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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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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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 ...

2
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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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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 $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...

3
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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 ...

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

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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 ...

4
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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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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 ...

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[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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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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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
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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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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?

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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(...

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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 $X$...

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$\DeclareMathOperator\Gr{Gr}$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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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 ...

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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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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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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 ...

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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 ...

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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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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 ...

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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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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^...

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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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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 ...

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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 ...

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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 ...