# Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

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### Finite ring extension of local rings, revisited

This question says the following: Let $R$ and $S$ be local rings with the maximal ideals $M$ and $N$, respectively. Assume that $R\subset S$ and that $S$ is a finitely generated $R$-module. If there ...
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### Resolving singularities via blowups

Let $X$ be a singular affine variety over a field $k$. I have learned that we can obtain a resolution of singularities (i.e. a nonsingular variety $W$ and a proper birational morphism $W \to X$) by ...
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### Rank of a quasicoherent sheaf $\mathscr{F}$

In Ravi Vakil (Foundations of Algebraic Geometry), $\S 13.7.4$, the rank of a quasicoherent sheaf $\mathscr{F}$ at a point $p$ of $X$ is defined as dim$_{k(p)}\mathscr{F}_p/m_p\mathscr{F}_p$. A first ...
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### A canonical equivariant structure on the structure sheaf: checking the cocycle condition

Let $G$ be a linear algebraic group, let $X$ be a $G$-variety (for simplicity, let $X$ be a complex affine variety, with associated structure sheaf $\mathcal O_X$, regarded as a sheaf of $\mathcal O_X$...
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### Maximal ideals in certain affine algebras

Let $a_1,\ldots,a_m \in \mathbb{C}[x_1,\ldots,x_n]$, with $m > n$, and let $R=\mathbb{C}[a_1,\ldots,a_m]$. For example, $R=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$. Question. Could we find all ...
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### geometric vector bundles associated to twisted structure sheaf

Let $V$ be a $k$-vector space with basis $v_1,\dots,v_n$. Let $G(1,V)$ be the Grassmannian of $1$-dimensional subspaces of $V$. In their book on intersection theory, in section 3.2.3, Eisenbud & ...
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### Using a lemma to calculate syzygy.

I want to find the syzygies of the following monomial ideal $I = (x_1^4, x_1^3x_2, x_1^2x_2^2, x_1x_2^3, x_2^4)$ in $S = k[x_1, x_2]$. To do this I will use Lemma 15.1 on pg. 322 in Eisenbud "...
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### When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

I am reading a set of notes by Michel Brion about automorphism groups of projective varieties. The following claim appears in the proof of a theorem stating that if G is a connected group scheme, $X$ ...
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### $k$-rational points of the automorphism functor of a scheme

Let $X$ be a scheme and let $\operatorname{Aut}_X$ denote the functor sending a scheme $T$ to the set of $T$-automorphisms of $X \times T$. Assume that $\operatorname{Aut}_X$ is representable by a ...
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### Examples/classification of algebraic symplectomorphisms

I'm curious about examples of algebraic automorphisms of complex varieties which are symplectomorphism. For instance, can we classify the algebraic symplectomorphisms of $\mathbb{P}_{\mathbb{C}}^n$ ...
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### Generalization of the finiteness of the class group for a projective scheme regular over $\mathbb{Z}$.

The finiteness of the class group, in schematic terms, means that if $K$ is a number field, then the Picard group of $\operatorname{Spec}\mathscr{O}_K$ is finite. I heard that it is true in general ...
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### Theorem about necessary and sufficient conditions for configuration of n-tuples of points.

I'm writing my thesis and at a certain point I claim, and indeed it turns out to be true from discussions with my supervisors, that certain configurations for n-tuples of points, in a curve of degree ...
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### Projective variety containing hyperplane at infinity

Let $V$ be a projective variety containing the hyperplane at infinity $H_{\infty}$. Why does it follow then that $V = \mathbb{P}^n$ or $V = H_{\infty}?$ Obviously $I(H_{\infty} )= (X_{n+1})$, but I ...
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### Isomorphism map keep smoothness?

Let $E$ be a smooth curve, that is, one dimensional projective variety with dimension one, and Jacobi matrix is non-singular at any point. And suppose that the map $φ$, which sends $E$ to $E'$, is ...
One construction of projective space over a ring $A$ is to take $n+1$ affine opens given by $$U_i = \operatorname{Spec} \frac{A\left[\frac{x_0}{x_i}, \dots, \frac{x_n}{x_i}\right]}{x_i/x_i-1},$$ and ...