# Tagged Questions

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

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### Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
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### Normalization of complete intersection

Let $A$ be an integral complete local ring over a field which is complete intersection. Let $B$ be a normalization of $A$. Q. Is $B$ Gorenstein? I guess that even the normalization of Gorenstein ...
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### Is the Grassmannian contained in a Plucker hyperplane?

Is the dimension of the projective space $\mathbb{P}^{{n \choose k} -1}$ into which we embed the Grassmannian $G(k,n)$ of $k$-planes in $n$-space minimal? In other words, is the Grassmannian variety (...
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### No cuspidal character sheaves on GL(n)

We need a reference for the fact that there are no cuspidal character sheaves on $GL_n$ unless $n=1$. See page 11 of http://www.kurims.kyoto-u.ac.jp/~arakawa/Henderson_mgsctalk2.pdf.
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### Invariance of the fiber-dimension of a finite map

Let $A\subseteq B$ be commutative Noetherian rings such that $A$ is a regular ring, i.e., $A_{\mathfrak{m}}$ is a regular local ring for all maximal ideals $\mathfrak{m}$ of $A$ and $B$ is a finite $A$...
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### What are Motivic homotopy types?

There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it. I would like to know the reference in which Grothendieck did it, ...
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### Clarification on the weak BSD conjecture

It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function $$f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}$$ as $x$ tends to ...
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### Psi-classes on moduli spaces of weighted curves

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...
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### Extending a prime divisor to a principal divisor

Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z_1, \ldots, Z_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist ...
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### Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem. Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...
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### Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
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### What is a field [Körper] really?

The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements: The ...
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### Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page. I am asking if it ...
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### K theory and derived categories

Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra". As next step, I moved to ...
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### Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$

I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), ...
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### Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say \...
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I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of $\... 1answer 213 views ### Normal bundle to fibers of a rational morphism Let$f:X\dashrightarrow C$be a rational fibration from a 3-dimensional variety$X$to a curve$C$such that generic fiber is smooth and different fibers intrsect in smooth curves. Take$S$to be a ... 0answers 133 views ### Stronger version of Bertini's theorem In char 0, is there a generalised version of Bertini's theorem that will ensure that for a proper map$f: Y\rightarrow X$between smooth projective varieties and for every point$x\in X$we can find ... 1answer 197 views ### Characters of simply connected semsimple algebraic groups over local fields Let$G$be a semisimple algebraic group over$\mathbb{Q}_p$. Then by definition$G$admits no non-trivial algebraic characters, i.e. homomorphisms$G \to \mathbb{G}_m$. However, it is quite possible ... 0answers 124 views ### Uniform bound on the Mordell-Weil rank of elliptic curves I want to know that if there is an uniform upper bound for the rank of elliptic curves over$\mathbb{C}(t)$, the rational function field over complex numbers, and generaly over the function field of ... 1answer 230 views ### j-invariants for isogenous elliptic curves Let$E$be a smooth complex elliptic curve, and$\sigma$translation of$E$given by a point$p$on$E$of finite order, with respect to some fixed origin. What are the$j$-invariants related with$E$... 1answer 147 views ### Methods of showing a variety is stably rational As anyone who follows the algebraic geometry tag on arXiv will probably know, there has been a lot of papers recently showing various varieties are non-stably rational. What I am interested in however ... 2answers 261 views ### Group cohomology of fundamental group of a curve The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this ... 0answers 218 views ### Construction of Dualizing sheaf I was going through the construction of dualizing sheaf given in Hartshorne [III, 7, Lemma 7.4]. The proof apparently omits lots of details. In particular it does not mention any of the$i^* , i_*, i^{...
Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$,...