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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.

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.

29074 questions
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Example II.$3.2.6$ in Hartshorne (reduced induced closed subscheme structure)

Consider a scheme $(X,\mathcal{O}_X)$ and a closed subset $Y$ of $X$. Let $X = \bigcup U_i, \, U_i \cong \operatorname{Spec}(A_i)$ be an open affine covering of $X$ and let $Y_i = U_i \cap Y$. Then $Y_i$ is closed in $U_i$ and it can be endowed with…
Manos
  • 25,833
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A polynomial of degree $k$ vanishing at $kd+1$ points on a rational normal curve in $\mathbb{P}^d$ must vanish on the whole curve

This is asserted in Exercise 1.15 of Joe Harris's algebraic geometry book (Algebraic Geometry: A First Course, Pg. 11 in my copy). This result struck my fancy but I'm unable to solve it myself or find references to it elsewhere. The closest…
Zach Conn
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What is the sheaf of differentials of projective space?

For any ring $A$, I wish to calculate the sheaf of differentials $\Omega_{\mathbb{P}_A^n/A}$. Let $S=A[x_0,\ldots,x_n]$, and let $S_{(x_i)}=A[x_{0/i},\ldots,\widehat{x_{i/i}},\ldots,x_{n/i}]$. I know that I have that…
Dominic Wynter
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Deformation to the normal cone

Fulton in his book "Intersection theory" uses local description of this deformation that I can't understand. I quote paragraph from page 87 and insert my questions. Assume $Y=\operatorname{Spec}(A)$, and $X$ is defined by the ideal $I$ in $A$. To…
Alex
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Serre Twisting Sheaf Etymology

Is there a reasonable and coherent explanation of why the Serre Twisting Sheaf has the word "twisting" in its name?
Tanner Strunk
  • 985
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Localizing a divisor on a scheme

Fix an integral Noetherian separated scheme $X$ that is regular in codimension 1, then let $D\in\operatorname{Div}X$ be any divisor, then Hartshorne claims that for any point $x\in X$, we can localize the divisor $D$ to obtain some divisor…
Dominic Wynter
  • 6,159
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Must a quasicoherent sheaf which is zero on a dense open subset be torsion?

Let $X$ be a scheme, and $F$ a quasi-coherent sheaf of $\mathcal{O}_X$-modules on $X$, such that for some dense open $U\subset X$, we have $F|_U = 0$. Under these assumptions, must $F$ be a torsion module? I'm happy to assume $X$ is noetherian and…
user355183
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Regarding a sheaf of $\mathcal O_X$-modules as a sheaf of $\mathcal O_Z$-modules, where $Z$ is a closed subscheme

Let $(X,\mathcal O_X)$ be a scheme, and $(Z,\mathcal O_Z)$ a closed subscheme, with $i: Z \rightarrow X$ the defining closed immersion. The underlying map of topological spaces is the inclusion. Since $i^{\#}: \mathcal O_X \rightarrow i_{\ast}…
D_S
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Separated varieties and uniqueness of extension

I would like to know if my solution to the following exercise is correct. If not, then I will be grateful for a correct argument. (I am working with varieties over an algebraically closed field, not schemes) Exercise: Suppose that $f,g:X \to Y$ are…
AKr
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Why are Prime Divisors on Curves just Closed Points?

Why is it true that prime divisors on nonsingular curves (curve: integral separated scheme of finite type over an algebraically closed field, of dimension 1 with all local rings regular) are only the closed points? In particular, if $C$ is curve,…
Manos
  • 25,833
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Connected component in a scheme

Let $X$ be a scheme over a field $k$ (if necessary, locally of finite type). Let $E$ be a connected component of $X$. Then $E$ is closed. For example, $X$ could be a group scheme over $k$, and $E = X^0$ the connected component of the identity…
D_S
  • 33,891
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Why is the Open ball in Euclidean topology not an affine algebraic variety?

Im slightly struggling to visualise the concept here. On the reals or the complex plane, the open ball with euclidean topology is not an affine algebraic variety. How is there no collection of polynomials that has it as its zero locus?
anoanoanoano
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An exercise 2.3 from Harris's AG: A First Course

This is an exercise 2.3 from Harris's AG: A First Course: Using the result of the exercise 2.2 ($A(\mathbb{A}^2 \setminus \{0\}) = K[x_1, x_2]$) show that $X = \mathbb{A}^n \setminus \{0\}$, $n > 1$, [as a quasi-affine variety] is not isomorphic…
Aleksei Averchenko
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Showing that $k^n$ is dense in $\mathbb{A}_k^n$

Let $k$ be an infinite field, not algebraically closed. Let $\operatorname{Specm} k[x_1,\cdots,x_n]$ be the set of maximal ideals of $k[x_1,\cdots,x_n]$ and define a topology on $\operatorname{Specm} k[x_1,\cdots,x_n]$ in which the closed sets are…
Manos
  • 25,833
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Blowing up families

Suppose that $g: X \to Y$ is a map of varieties (perhaps flat), and we have a subvariety $V \subset X$. Important: I want $V$ to be flat over $X$, otherwise there are trivial counter examples. Consider now the $f : Bl_V X \to Y$ as a composition…
Elle Najt
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