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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10 views

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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Is an irreducible affine variety isomorphic to some affine hypersurface?

I'm reading the basics of birational geometry in Shafarevich's "Basic Algebraic Geometry, 1" third edition. In theorem 1.8 he proves that Every irreducible affine variety $X\subseteq \...
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A surjective morphism of complex algebraic varieties with smooth fibers that is a submersion

Let $f\colon X\to Y$ be a surjective morphism of irreducible complex algebraic varieties with nonsingular fibers. Moreover, assume that each geometric fiber $f^{-1}(y)$ is isomorphic to an affine ...
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25 views

Ravi Vakil's notation $\operatorname{Proj}_A S$

I saw the notation $\operatorname{Proj}_A S$ for a ring $A$ and a graded $\mathbb{Z}^{\geq 0}$-graded ring $S$ being used in Ravi Vakil's notes on Algebraic Geometry. I first came across the notation ...
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Dimension of irreducible affine set is same as any open subset

I'm partially studying the book Algebraic geometry of Daniel Perrin, and I have a doubt on proposition 1.11 of chapter 4. For some reasons I'm reading this section without reading the previous one ...
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16 views

What are the R - families of connections

I've the definition of a connection on a vector bundle and on a G-principal bundle. But reading I have found this statement if R is a $\mathbb{C}-$algebra take the R-families of connections over the ...
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Solution to equations v.s. submanifold

Consider a manifold $M$, define a subset $A$ in $M^{2n}$ for some $n$ $$A=\{(x_1,x_2,...,x_{2n})\in M^{2n}| x_{2i-1}=x_{2i}\ \text{for} \ i=1,...n \}$$ Is $A$ necissarily a submanifold? In general, I ...
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Open affine subscheme of affine scheme with different underlying set is non-isomorphic

If I have an affine scheme $X=\operatorname{Spec}A$, and an open subscheme $Y=\operatorname{Spec}B$ which is also affine and whose topological space is different from $X$, does it follow that $X$ and $...
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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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Is the formal spectrum locally noetherian?

I am studying the formal spectrum and I came up with a really nice example, that is, the ring of formal power series $\text{Spf}(k[[t]])$. My question is, what are the main properties of the space $\...
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Two seemingly non-isomorphic elliptic curves over a finite field which have the same cardinality

Let $p$ be a prime number such that $p \equiv 3 \mod 4$. Now consider the elliptic curves $$ E/\mathbb{F}_{p^2}: \quad y^2 = x^3 - ax \quad \text{and} \quad E'/\mathbb{F}_{p^2}: \quad y^2 = x^3 - a^{-...
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Flat morphism of local rings

Let $(A,m_A)$ and $(B,m_m)$ be two Noetherian local rings, $A \subseteq B$ and $B$ is a finitely generated $A$-algebra. Step 1: Assume that: (1) $A$ is regular. (2) $A \subseteq B$ is flat. Question ...
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Recovering the cohomology of non-locally constant sheaves on anabelian schemes.

Connected affine schemes in char $p$ are $K(\pi_1,1)$, it is expected that in the aforementioned case the schemes are anabelian (These are stated in this). I was wondering since it is expected to be ...
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What are the initial and terminal objects in the category of sheaves of abelian groups?

What are the initial and terminal objects in the category of sheaves of abelian groups? (I know here bpth initial and terminal objects are the same since it is an abelian category.... but I need more ...
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Is the theory of sheaves of abelian groups a first order theory? [closed]

I need to know if the theory of sheaves of abelian groups a first order theory? Are there aarbitrary large models for this theory exist?
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Why should we study morphism of $k$-schemes instead of morphisms of schemes?

I have been self-studying algebraic geometry through youtube video lecture of Prof. Richard E. BORCHERDS. This is the link of the lecture that I have question about: https://www.youtube.com/watch?v=...
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Can we choose constants such that a system of two polynomials in two variables has finitely many solutions?

Given $f(x,y)$ and $g(x,y)$ be polynomials with rational coefficients does there exist $h, k \in \mathbb{R}$ such that the system: $$ f(x,y) = h $$ $$ g(x,y) = k $$ has finitely many solutions $(x,y)$....
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Why is the functor $S\mapsto$ isomorphism classes of curves of genus $g$ over $S$ not representable?

Whenever $R_{1}\hookrightarrow R_{2}$ is an injection of rings, \begin{equation*} \operatorname{Hom}(R,R_{1})\hookrightarrow\operatorname{Hom}(R,R_{2}) \end{equation*} is an injection for any ring ...
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Trace morphism in Lipman's “Dualizing sheaves, differentials and residues on algebraic varieties”. [duplicate]

Let $f: V \to W$ be a finite surjective morphism of varieties, with $W$ proper and normal. Does there exist a trace map $$ \operatorname{trace}\colon f_* \mathcal O_V \to \mathcal O_W? $$ I know that ...
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Degree of a subvariety of the Grassmannian

This might be a stupid question due to my lack of knowledge in algebraic geometry. So I have a subvariety $V\subset G_k(\mathbb{C}^n)$ of codimension $k$. The only thing I know about $V$ is its ...
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Determine the splitting type of a bundle with cokernel as $\mathcal{O}_2p$ or $\mathcal{O}_p\oplus\mathcal{O}_q$

In my situation, I want to determine the splitting type of a rank 2 vector bundle $\mathcal{F}$ fitted in an exact sequence $$0\rightarrow\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(-2)\...
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62 views

How come the genus of algebraic curve can be any natural number?

One the one hand, any topological surface has a complex structure, so for any natural number $g$, there exists a complex curve with genus $g$. On the other hand, we have a Chow's theorem, which says ...
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1answer
36 views

Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project. First, Modules Definition 8.1 states that a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is locally generated by sections if for all ...
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34 views

Dimension of a scheme $X$ at a closed point $x$ and dimension of its local ring.

The dimension of an irreducible scheme $X$ at $x$, dim$_x(X)$ is defined as the smallest dimension among its open neighbourhoods and the dimension of its local ring dim$(\mathcal{O}_{X,x})$ is just ...
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54 views

Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$. Assume that: (1) $R$ and $S$ are (Noetherian) integral domains. (2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
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49 views

The point $a \in X$ is smooth then there is only one irreducible component of $X$ meeting $a$.

I am reading Gathmann's note on Algebraic Geometry and there is a remark that I really don't understand in the chapter 10. First I recall a result that is, apprently, needed to understand the remark. ...
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58 views

Showing that $\{g_1, \dots , g_s\}$ is also a Gröbner basis for $M.$

Let $F$ be a free module (of finite rank) over $S = k[x_1, \dots , x_r]$ with monomial order >. Let $M \subset F$ be a submodule and let $B = \{g_1, \dots , g_t\}$ be a Gröbner basis for $M.$ I ...
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60 views

Zariski topology is always strictly finer than product topology

Hi know there are similar question but i haven't found an answer to this particular one. Given an infinite Field $k$, show that for any n,m strictly positive integers the zariski topology on $k^{m+n}$ ...
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Stratification of the affine grassmannian for $G = GL_2$

Apologies for a very stupid question. I am trying to understand closure of strata in the complex affine grassmannian for $GL_2$. Recall that $Gr = GL_2(K)/GL_2(O)$, where $K = \mathbb C((t))$ and $O = ...
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1answer
28 views

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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1answer
45 views

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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78 views

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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The Hilbert function and polynomial of $S = k[x_1, x_2, x_3, x_4]$ and $I = (x_1x_3, x_1x_4, x_2x_4)$ step clarification.

My professor based on pg. 320 - 321 of Eisenbud, wrote the following: Let $I = (m_1, \dots, m_l)$ be a minimal set of monomial generators, $I' = (m_1, \dots, m_{l-1}) \subsetneq I,$ and $d = \...
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Is there always a toric resolution between a toric variety and a desingularization?

My question is related to this MO question, where "resolution" is replaced by "isomorphism". Let $X,Y$ be projective toric varieties, $X$ smooth. Suppose there exists a proper ...
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1answer
64 views

Maps between shifted complexes

I am studying the higher stack of perfect complexes $E$ of fixed length $n=(b-a)+1$ (i.e. their homology groups are zero outside the interval $[a,b]$) and it seems that the higher homotopy groups (...
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1answer
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Finding the syzygy (relation module) of a monomial ideal.

I have read pages 322-323 of "Commutative algebra, with a view toward algebraic geometry" by David Eisenbud, but it is still not much clear what are the steps of finding the syzygy. I am ...
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1answer
75 views

The tangent space to an affine variety has a “local formulation”.

I am studying the chapter 10 (Smooth Varieties) in Gathmann's lecture notes on Algebraic Geometry and there are several things that I don't understand. Let me recall the context. He defined the ...
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1answer
77 views

Understanding some details in calculating the Hilbert function and polynomial.

I am practicing finding the Hilbert function and Hilbert polynomial for $S/I$ of the following: $S = k[x_1, x_2, x_3, x_4]$ and the monomial ideal $I = (x_1 x_2 x_3, x_2x_3x_4, x_1x_4).$ Note that, I ...
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1answer
52 views

If two varieties are isomorphic so is their blow-up.

I'm following Gathmann notes on Algebraic Geometry. In Problem 10.17, we are to prove the affine curves $X_k = V(x_2^2 - x_1^{2k+1})\subset \mathbb{A}^2_{\mathbb{C}}$ are not isomorphic for different $...
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156 views

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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1answer
88 views

$k$-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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1answer
38 views

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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67 views

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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21 views

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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1answer
17 views

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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1answer
54 views

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 ...
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1answer
29 views

Projective space as the glue of affine schemes: checking the cocycle condition

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 ...
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1answer
76 views

Zariski site is subcanonical?

I want to show that the zariski site is subcanonical as an exersice of the book "Sheaves in geometry and logic" and I need help with it... To be honest I didnot really understand the ...

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