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        Stack Exchange Network

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        for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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        40 views

        Squarefree monomial ideals, locally complete intersections

        Let $I\subset S=\mathbb{C}[x_0,\dots,x_n]$ be an ideal generated by squarefree monomials of the same degree. Is it true that $S/I$ is a locally complete intersection?
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        101 views

        Convergent sequences in projective varieties

        It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...
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        176 views

        What is the mirror of an algebraic group?

        Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
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        1answer
        141 views

        Some questions about Clemens' paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory

        I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the ...
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        0answers
        78 views

        Holomorphic map, Instantons of Complex Projective Space and Loop Group

        It seems that holomorphic (or rational) maps play a crucial role to relate the following data: Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime. ...
        4
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        91 views

        The geometry and arithmetic of the intersection of a cubic and quadric threefold

        Let $f,g \in \mathbb{Z}[x_0, \cdots, x_4]$ be a quadratic and cubic form (i.e., homogeneous polynomials) respectively, and let $V(f), V(g)$ denote their respective projective varieties; $V(f), V(g)$ ...
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        1answer
        141 views

        A possible “finite set avoiding” version of Grothendieck's Zariski Main Theorem?

        Recall that in EGA $IV_3$ (Théorème (8.12.6)), Grothendieck calls the following theorem ``Zariski's Main Theorem": Let $Y$ be a quasi-compact separated scheme, and $f : X \to Y$ is a separated ...
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        0answers
        113 views

        Can free rational curves lift to ramified covers of Fano varieties?

        Does there exist $X$ a smooth Fano manifold, $f: Y \to X$ a nontrivial ramified finite cover, $C \subseteq X$ a smooth very free rational curve, such that $f$ is étale over a neighborhood of $C$? ...
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        1answer
        682 views

        Does every sheaf embed into a quasicoherent sheaf?

        Question. Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?...
        5
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        0answers
        137 views

        Infinite-dimensional manifold $ΩG$ of loops and the complex projective $n$-space

        In a review seminar today, I heard the speaker takes the below for granted, but I have no enough background "Yang-Mills (YM) instantons in 4D can be naturally identified with (i.e. have the same ...
        5
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        2answers
        131 views

        Degree of the variety of independent matrices of rank $\leq r$?

        Consider an $m$-by-$n$ matrix $A$ with entries in a field $k$; we can see $A$ as a point in the affine space $\mathbb{A}^{m n}$. The rank of $A$ will be $\leq r$ (where $r<\min(m,n)$) if and only ...
        3
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        0answers
        77 views

        Periods for Irreducible Holomorphic Symplectic Manifolds

        Let $f:\mathscr{X}\rightarrow \operatorname{Def(X)}$ be the Kuranishi family of $X$, where $X$ is an irreducible holomorphic symplectic manifold. After shrinking $\operatorname{Def}(X)$, we get that ...
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        0answers
        37 views

        Geometric intuition of the dimension of Grassmannians and flag manfolds [migrated]

        I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{...
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        0answers
        83 views

        singularity of an irreducible surface in $\mathbb{P}^3$

        Is it true that the singular locus of an irreducible hypersurface in $\mathbb{P}^3$ have pure co-dimension ?
        2
        votes
        1answer
        153 views

        Computation of cohomology of ideal sheaves

        Let $j: X \to Y$ be a closed embedding. Let $I_{X/Y}$ be the ideal sheaf of this closed embedding. Then there is a exact sequence $$ I_{X/Y} \to \mathcal{O}_Y \to j_{*}\mathcal{O}_X \to 0$$ One use ...

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