# Questions tagged [ag.algebraic-geometry]

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

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### Schur polynomial with integer values

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 ...
1 vote
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### Examples of smooth compact Kähler manifolds with semipositive canonical class

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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### Filtration of Chow variety based on the dimension of the intersection of cycles

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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### Boolean operation on n dimensional polyhedron

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 ...
112 views

### Trace formula for monodromy of Milnor fibrations

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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### Global generation of $S^n \Omega_X$ for a fake projective plane

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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### Every locally free sheaf is Cohen-Macaulay on complex variety with at canonical singualities?

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) ...
• 191
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### Right adjoint of subcollection of semi-orthogonal decomposition

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 ...
148 views

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1 vote
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### Hodge autoisometry and Hodge automorphisms of a hypersurface

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 ...
136 views

### A map between Brauer groups

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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### PAC and totally real fields

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 ...
251 views

### Higher-order HKR theorems?

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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1 vote
192 views

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### Locus of points with reducible fibers over moment map

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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### What is the natural linearization on differentials?

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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### Restricting perverse intermediate extension to closed complement

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 ...
• 325
1 vote
71 views

### In complex analytic category, is the pluricanonical sheaf Cohen--Macaulay?

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 ...
• 191
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### On the noetherianess of some subalgebras of an affinoid algebra

$\DeclareMathOperator\Sp{Sp}$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 ...
60 views

### Non-tree models of Lagrange inversion polynomials

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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### Künneth formula for $\pi_1$-proper morphisms

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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1 vote
164 views

### Lefschetz theorem on (1,1) classes for a compact complex surface

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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### Nullstellensatz with nilpotents and $I=J(V(I))$

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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