Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
23,109 questions
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Characteristic stalks of log curves and their log modifications: an apparent contradiction
This is about log curves in log algebraic geometry and also involves twisted curves as in Olsson's paper (log) twisted curves.
I will give the setup and then provide an argument justifying every step ...
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Extending the "infinitesimal automorphism" construction of derivations to algebraic geometry
In the context of smooth manifolds $M$, there is a well-known correspondence between the "infinitesimal" versions of certain objects and derivations of the algebra $C^\infty(M, \mathbb{R})$. ...
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On Hessian matrix of any local artinian Gorenstein ring
Let $k$ be a field. For a standard graded Artinian Gorenstein $k$-algebra $A = S/I$ where $S = k[\![x_1,\ldots,x_n]\!]$, the Macaulay inverse system identifies $A$ with a homogeneous polynomial $F \in ...
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Analytic de Rham stacks of Fargues-Fontaine curves?
Earlier this year, Johannes Anschütz, Arthur César Le Bras, Guido Bosco, Juan Esteban Rodríguez Camargo and Peter Scholze published a preprint titled “Analytic de Rham stacks of Fargues-Fontaine ...
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Irreducibility of fibres of dominant morphism with equidimensional fibres
Let $f: X \to Y =\operatorname{Spec}(A)$ a dominant morphism between irreducible smooth varieties (= $k$-schemes of finite type) such that
the generic fibre $X_{\eta}$ is irreducible
every fibre $X_y$...
- 3,952
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How to write down a crepant resolution of this variety?
We work over $\mathbb{C}$. I have been studying crepant resolutions of threefolds and am trying to write down explicit examples. Let $X\subset \mathbb{A}^{4}$ be the hypersurface defined by $y^{2}=x^{...
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Grothendiek duality and dualizing sheaf
Let $i:Z\to X$ be a closed immersion and quasi-smooth map between derived algebraic stacks. Let $\omega$ be the dualizing sheaf on $X$. Then is the following isomorphism true? Here $i^!$ is the right ...
- 1,191
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Quasi-split and inner forms of reductive groups over a curve
Let $X$ be a curve over an algebraically closed field and let $G$ be a split reductive group. Over the generic point $\eta$, Steinberg's theorem implies that we have exact sequence
$$
0 \to H^1(\eta, \...
- 846
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Derived pullback between locally Noetherian schemes preserves bounded above complexes with coherent cohomology
$\def\Dmc{D^-_\mathrm{coh}}$Let $f:X\to Y$ be a map of locally Noetherian schemes. Is it true that $Lf^*\Dmc(Y)\subset \Dmc(X)$? When $X$ and $Y$ are Noetherian this follows by [GW, 21.157, 22.61] (in ...
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Algebraization of the Tate curve
Let $A$ be a Noetherian ring which is $t$-complete for some $t\in A$. In [1], the formal scheme $\hat C$ is defined by
$$C_N=\overline{\mathscr{G}}_\mathrm{m}^t/g^{\mathbb Z}$$
over $A/(t^N)$ for $N\...
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Finding rational points to an elliptic curve equation with very large coefficients
I have been working on this elliptic curve equation:
$y^2 = 102516622445224071181618629075661543598435443927642055302797028500\cdot x^{3} + ...
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Descent theory and global sections
Let $g: S' \to S$ be a quasi-compact faithfully flat morphism and let
$\text{pr}_i : S' \times_S S' \to S'$ & $\text{pr}_{ij} : S' \times_S S' \times_S S' \to S' \times_S S'$ ($i=1,2,3$) the ...
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Is the resolution of Kato's exotic nilpotent cone symplectic?
S. Kato introduced the exotic nilpotent cone and its resolution of singularities in his papers such as "An exotic Deligne-Langlands correspondence for symplectic groups".
My question is: is ...
- 3,504
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Half-integer Tate twist in the analytic geometric Satake equivalence
Let $G$ be a split reductive group over a (suitable) field $k$, the geometric Satakeequivalence of Mirkovic-Vilonen states that there is an equivalence of Tannakian categories
$$\operatorname{Perv}(\...
- 1,275
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How do I remove the topologically-nilpotent hypothesis from K-L's proof of equidimensionality of affine Springer fibers?
Let $G$ be simply connected semisimple over local field $F$. In Kazhdan--Lusztig's foundational paper on affine Springer fibers, they assume throughout that $\gamma \in \mathfrak{g}(F)$ is ...
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