# 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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There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...

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Suppose $(M, \omega)$ is a Kähler manifold, and I am looking for examples of compact Kähler manifolds with $c_1(K_{M}) \geq 0$. A $(1,1)$ form $\eta$ is semi-positive if in local coordinates its ...

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Let $X$ be a complex projective variety. Let $Y$ be subvariety of $X$. Consider the Chow variety of $r$ cycles on $X$ denoted by $C_r(X)$ (We can assume it is the Chow of variety of irreducible cycles)...

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A polyhedron in $R^n$ is defined by a set of half-planes: $P = \{x \in R^n \mid Ax - b \le 0\}$.
Given a set of polyhedra in $R^n$, $ P_1, P_2, \dotsc, P_k$, is there an algorithm/implementation that ...

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I am reading the paper A. Campo, Le nombre de Lefschetz d'une monodromie but I am stuck at several points, hope that someone can help me.
Let $P:\mathbb{C}^{n+1} \longrightarrow \mathbb{C}$ be a germ ...

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Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...

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Suppose $X$ is a normal complex space with at most singularities, can we say any locally free sheaf on it is Cohen-Macauly?
Recall that a coherent sheaf $\mathcal{F}$ over $X$ is called (maximal) ...

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Suppose $X$ is a prime Fano threefold of index 1 such that $H = -K_X$ is ample. There is a full classification of the derived category of such threefolds depending on the genus of $X$; in the case ...

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I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...

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Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...

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Consider a complex projective hypersurface $X\subset\mathbb{P}^{n+1}$, then we have the natural Hodge structure on the middle cohomology $H^n(X,\mathbb{Z})$. In which cases (examples) does the group ...

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Let $R$ be a henselian dvr over $\mathbb{C}$ and $A$ be a flat $R$-algebra of finite type. Suppose $\hat{R}$ is the completion of $R$ and $\hat{A}:=A\otimes_R \hat{R}$. For an ideal $I\subseteq A$, ...

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A field $K$ is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety over $K$ has a $K$-point. Let $L$ be the maximal totally real subfield of $\overline{\mathbb Q}$. A few ...

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Recall that Hochschild-Kostant-Rosenberg -type theorems identify certain smoothness conditions guaranteeing an isomorphism between the cotangent complex and (a shift of) the Hochschild homology of an ...

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Consider a smooth projective variety of ample $\omega_X$, how can I quickly see that
$$\textbf{Coh}(X)=\{\mathcal{F}^{\bullet}\mid\text{Hom}(\omega_X^{\otimes i},\mathcal{F}^\bullet[n])=0\text{ for } ...

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Suppose $G$ is a locally free sheaf on $P^{d}$. $F_{1}$,$F_{2}$ are two subsheaves of
$G$ and they concide on a dense open subscheme of $P^{d}$ .If the quotients of $G$ corresponding to these two ...

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In Huybrechts and Lehn's book "The Geometry of Moduli Space of Sheaves", a sheaf $E \in Ob(Coh_{d,d-1}(X))$ is polystable if $E \cong \bigoplus E_{i}$ in $Coh_{d,d-1}(X)$, where the
sheaves $...

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I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand ...

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Let $f:X\rightarrow W, g:Y\rightarrow W$ be projective surjective morphism between normal projective varieties, do we always have $\omega_{X\times_W Y/W}=f^*\omega_{X/W} \otimes g^* \omega_{Y/W}$? ...

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Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...

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Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...

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Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...

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Is a manifold with corner an analytic space (just show that $\left[0, +\infty \right)^{n}$ is an analytic space, which seems obvious but maybe I'm wrong...) EDIT: as noted in the comments some complex ...

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Let $n\geq 4$ and $X\subset \mathbb{P}^n$ be a smooth hypersurface. Let $p(t)\in \mathbb{Q}$ be a polynomial and we consider the Hilbert scheme $Hilb^{p(t)}(X)$.
If a point $[Y]\in Hilb^{p(t)}(X)$ ...

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$\DeclareMathOperator\GL{GL}$Let $ \overline{\mathbb{Q}}_{p} $ be an algebraic closure of $ p $-adic numbers $ \mathbb{Q}_{p} $. A closed subgroup $ H $ of a general linear group $ \GL_{n}(\overline{\...

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Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:
Question 1. Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan ...

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Suppose $X$ and $Y$ are smooth affine surfaces over $\mathbb C$. Suppose there is a biholomorphism $f: X\to Y$. Does it follow that $X$ and $Y$ are isomorphic as affine surfaces (i.e. there exists an ...

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When computing the dimension of moduli space for complete intersections of type $(a,b)$ in $\mathbb{P}^n$, what do we need to consider? In general we have the following part:
$$|\mathcal{O}_{\mathbb{P}...

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Suppose $X$ is a Fano manifold of Picard rank one such that the cotangent bundle $T^*X$ is completely algebraically integrable system. Let $\mu: T^*X \to \mathbb{C}^n$ be the moment map, where $n$ is ...

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Suppose that $R$ is a complete DVR with characteristic 0 fraction field $K$, maximal ideal generated by $p$ and characteristic $p>0$ residue field $k$ which is algebraically closed. Suppose that $\...

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Let $X$ be a smooth projective variety over $\mathbb{C}$ with $\dim X=3$ and $\mathrm{Pic}(X)=\mathbb{Z}\cdot D$, where $D$ is a very ample effective Cartier divisor on $X$. Let $Z$ and $C$ be two ...

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In Section 9.5 of Fangyang Zheng's Complex Differential Geometry Book, he proves the following:
Lemma 9.25. Let $(M^2,h)$ be a Kähler surface and $p \in M$. Suppose $M$ has negative holomorphic ...

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Let $X$ be a regular, projective flat scheme over $\Bbb{Z}$, let $\bar{L}$ be a hermitian line bundle on $X$. In order to define the height of an integral closed subset $Y$ we define it on closed ...

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This question is related to the answer of this post by abx: Curves and semi-abelian varieties.
Let $X$ be a smooth projective curve, and let $C$ be the nodal curve obtained by identifying the points $...

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Let $\Bbbk$ be a field. Let $G$ be an affine algebraic group over $\Bbbk$. Let $X$ be a scheme over $\Bbbk$. Let $G$ act on $X$ with the action morphism $\sigma:G\times X\to X$. There are two ...

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Consider a scheme $X$ over athe complex numbers, $j:U\to X$ an open subscheme, $i:Z\to X$ its closed complement, and a perverse sheaf $F$ over $U$ with complex coefficients.
The intermediate extension ...

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We adopt the following definition of canonical singularities in complex analytic category.
Let $X$ be a normal complex space of dimension $n$, and let $j:X_{\text{reg}}\rightarrow X$ be the open ...

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$\DeclareMathOperator\Sp{Sp}$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 ...

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The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...

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Let $R$ be a henselian DVR with fraction field $K$ and residue field $k$ of characteristic $p>0$. Let $\overline K$ be an algebraic closure of $K$, $\overline R$ the normalization of $R$ in $\...

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Given a set of rational points $S$, does there always exist a hyperelliptic curve $C$ such that $C(\mathbb{Q})=S$?
Namely, which sets could arise as the set of rational points of a hyperelliptic curve?...

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Geometry is not my area, and so, I am not sure the title accurately captures what I am interested in exactly... I hope the tags are appropriate.
For any vector $v$, denote it's $i$-th component by $v_{...

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Context:
Let $X$ and $Y$ be connected qcqs schemes over an algebraically closed field $k$. Denote by $\pi_1(X)$, $\pi_1(Y)$ their étale fundamental groups (base points omitted). Grothendieck proved ...

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In Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius' book Compact complex surfaces. Second edition p.142, there is a Lefschetz theorem on (1,1) classes for compact surface:
...

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Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero ...

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Let $X \subset \mathbb{P}^N$ be a smooth Fano variety of dimension $n \ge 5$ and $U$ be an open subset of $X$ whose complement has co-dimension at least $4$. Let $Y$ be the intersection of $X$ with $...

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Let $(R,\mathfrak m)$ be a local Cohen-Macaulay reduced ring of dimension at least $2$. Then, can we find a non-zero-divisor $x\in \mathfrak m$ such that $R/xR$ is again a reduced ring?
If needed, I ...

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Let $X$ be a complex projective variety, let $Y$ be a fixed subvariety. Consider all effective irreducible algebraic $r$-cycles where $r$ is fixed and chosen in a way such that $r \ll \text{Codim}_X(Y)...

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This may be a very silly question but I could not get any counter-example.
Let $M$ be a compact differential $4$-manifold with boundary $dM$.
Suppose that the inclusion map induced map $\pi_1(dM) \to \...

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Let $i_1: X_1 \to Y$ and $i_2: X_2 \to Y$ be two monics maps of schemes flat and locally of finite presentation over $S$.
Assume that for all $s \in S$, the fibral map $i_1(s)$ factorizes through $i_2(...