# Questions tagged [ag.algebraic-geometry]

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

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• 483
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### Is there a name for the number $n$ of points in general position s.t. $\operatorname{Cox}(\operatorname{Bl}_{p_{1},\dots,p_{n}}(Z))$ is not f.g.?

Let $Z$ be a projective, normal, $\mathbb{Q}$-factorial variety (so the Cox ring of $Z$ is well defined). Is there a name in the literature for the minimal natural number $n$ such that the ...
• 483
66 views

### Derived subgroup of rational points vs. rational points of derived subgroups

Let $G$ be a connected split reductive group over a field $k$. In general, we have an inclusion $$f: [G(k), G(k)] \rightarrow [G,G](k).$$ If $k$ is not algebraically closed, $f$ is not necessarily ...
• 2,221
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### Converting an iterative function to calculate the trajectory of a particle, into a polar function

I have been working on trying to convert the below code (which simulates the trajectory of a particle [with air friction]) into a polar function that can be evaluated immediately (without having the ...
• 1
1 vote
97 views

### what is the linear system on a cubic surface giving the blow-down map to the plane

Consider $X$ a smooth cubic surface in $\mathbb{P}^3$, and let $l_1,...,l_6$ be six disjoint lines contained in $X$. What is the linear system giving the blow-down map $X \to \mathbb{P}^2$, so that ...
• 412
1 vote
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1 vote
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• 483
233 views
+50

### Construction of a line bundle from a class $[\alpha] \in H^1(X, \mathcal{O}_X^{\times})$ as $\mathcal{O}_X^{\times}$-Torsor

Let $X$ be a complex compact manifold, and write $\mathcal{O}_X$ for the sheaf of holomorphic functions on $X$. Let $\mathcal{O}_X^{\times}$ be the subsheaf consisting of holomorphic functions. These ...
• 181
1 vote
243 views

### Sheaf cohomology in number theory

I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
• 123
146 views

### Approximate versions of Segre's Theorem

Consider projective $2$-space over a finite field of odd prime characteristic $p$. We say a set of points, $A$, in this space is an arc if any line meets it in at most two points. We say that an arc ...
• 11.4k
1 vote
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### Bundles vs. line bundles

Let $K$ be an algebraically closed field and consider the category $\text{Bun}$ of (finite dimensional) vector bundles over a $K$-variety $X$. Consider also the category of $\mathbb{G}^\times$-...
• 4,493
1 vote
212 views

I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions. Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
• 13
1 vote
106 views

### The Brauer group of the function field of a proper curve

Let $X$ be a smooth proper geometrically connected curve over a number field $k$, and let $k(X)$ denote its field of rational functions, i.e., its function field. Then the (cohomological) Brauer group ...
• 701
142 views

### References to let me know about current directions of research in arithmetic geometry

I have knowledge of basic algebraic geometry and good deal of number theory. I have studied roth theorem and I am currently studying proof of Mordell-Weil theorem. These two topics come under ...
• 572
248 views

### The Grothendieck topology of closed immersions on schemes

Let $S$ be a scheme. Let's define a Grothendick topology on $\mathrm{Sch}/S$ where a covering family $\{f_i:Z_i\rightarrow X\}_{i\in I}$ on an $S$-scheme $X$ is a collection of closed immersions of $S$...
• 279
1 vote
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### Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
• 429
1 vote
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• 1,656
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### Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism

Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...
• 1,686
150 views

### How the automorphism group of an elliptic curve acts at the localization of the stack $\mathcal{M}_{1, 1, k}$ at the corresponding point

I am studying the enlightening article "The Picard Group of $\mathcal{M}_{1, 1, S}$", written by Fulton and Olsson, but I have some problems with a proof. Setting Let $\mathcal{M}_{1, 1, k}$ ...
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59 views

1 vote
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### Is the Schofield semi invariant defined at $V/IV$?

Let $A=\mathbb{K}Q$ be the path algebra of an acyclic quiver $Q$ over an algebraically closed field $\mathbb{K}$, and $0\not=I\subset\mathbb{K}Q$ be an admissible ideal. Let $W$ be a left $A$-module ...
• 533
118 views

### Cone of morphism induced by Serre duality

For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category : $$S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]$$ ...
139 views

### Normal bundle of a linear subspace

Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$. If $\dim(H) = 1$, that is $H$ ...
• 307
161 views

### Are there any non-elementary functions that are computable?

Does a function $\mathit{f}:\mathbb{R}→\mathbb{R}$ being non-elementary (not expressible as a combination of finitely many elementary operations), imply that it is not computable? The particular case ...
• 11
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### Pull back of ample divisor by a birational morphism

Let $X$ be a normal projective variety of Picard number one. Let $\pi: Y \to X$ be a resolution of singularity of $X$ and $A$ is the ample generator of $\text{Pic}(X)$. Then $\pi^*A$ is nef but not ...
• 467
175 views

### Ample toric divisors

Let $X$ be a smooth complex projective variety (of dimension two or higher) and $D=\bigcup D_i$ be a simple normal crossings divisor on $X$ such that: $D$ is ample (maybe we need very ample but I am ...
130 views

### Find an analogue of Weyl chamber structure

Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma_1,..., \gamma_l$ of the weight lattice $X(T)$ such that the cone $C$ in ...
53 views

### Semistability of the pull back of sheaf of logarithmic one-forms modulo the torsion under birational map

Let $X$ be a projective Fano variety ( need not be smooth) of Picard number one. Let $D$ be a reduced divisor on $X$ and $U$ be an open subset contained in smooth locus of $X$ such that $D \cap U$ is ...
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• 133
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### Compatibility conditions for quadratic equations

In the context of physics, I stumbled over the following problem: I have $N$ equations, all are quadratic in a single scalar, real variable $x$: \begin{eqnarray} 0 &= A_1x^2 + B_1x + C_1 \\ &...
67 views

### Intersection product of $\mathbb{Q}$-Cartier divisors with irreducible complete curves is well-defined

I am learning the notion of intersection product of a $\mathbb{Q}$-Cartier divisor with an irreducible complete curve on a normal variety. The definition I learned is that if $D$ is a $\mathbb{Q}$-...
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