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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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Mumford's regularity theorem

I am reading the section on Castelnuovo-Mumford regularity from Lazarsfeld's Positivity in algebraic geometry. Theorem 1.8.3 in the book reads as follows: Let $F$ be an $m$-regular sheaf on $P^n$. Then for every $k \geq 0$ (i). $F(m+k)$ is generated…
user52991
  • 708
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trisecant lemma (reference + result over the real field )

I found the following result in an unpublished lectures notes Let $\mathbb X\subset P^N$ be a non degenerate subvariety of codimension $l>k$. Then the general $(k+1)$-secant $\mathbb P^k$ is not $(k+2)$-secant. The result was called $the$…
Ignat Domanov
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Geometric similarities between points in an algebraic variety

If $f : \mathbb{R} \to \mathbb{R}$ is a univariate irreducible polynomial, Galois theory says that all roots are equivalent up to field automorphism (specifically, an automorphism of the field extension fixing the base field). Can anything similar…
Geoffrey Irving
  • 858
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What is an example of an etale morphism which does not locally factor as a composition of an open immersion and a finite etale map?

Let $f:X\rightarrow Y$ be an etale morphism of schemes. By $f$ factoring locally as a composition of an open immersion and a finite etale map, I mean that for every $x \in X$ there exists an open $U$ of $x$ and an open $V \subset Y$ containing…
xlord
  • 285
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Quasicompact over affine scheme

Let $X$ be a scheme and $f : X \rightarrow \mathrm{Spec}\, A$ a quasicompact morphism. Are there any easy conditions on $A$ under which we can say that $X$ is quasicompact? Quasicompact morphism means only that there is an affine cover $\cup_{i \in…
NotfromBrazil
  • 237
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How many smoothings are there for a nodal curve?

Let $X_0$ be a projective nodal curve. It is known that one can find a smoothing of $X_0$: a family of projective curves $\pi:X\to B$ over a regular curve $B$, which is a smooth morphism over $B\setminus\{b_0\}$, and such that $X_0$ is isomorphic to…
Brenin
  • 14,072
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What is the definition of "local equation(s)" for a subscheme?

Hartshorne mentions "local equations" a few times without (so far as I can tell) actually defining them anywhere. As best as I can guess, the definition would be something like this: If $Y \subseteq X$ is a closed subscheme, then "local equations…
Daniel McLaury
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Why can you check closedness on the fibers? (Hartshorne's proof of Bertini's theorem)

In Hartshorne's proof of Bertini's theorem, given a linear system $|H|$, he defines the locus of "bad" hyperplanes $B_x$ for each point $x\in X\subset \Bbb P^n$ a projective variety, shows that this is a proper linear subset of $\{x\}\times |H|$,…
Hank Scorpio
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Proposition 6.6 in Hartshorne II: Bijectivity

Proposition 6.6 in Hartshorne II seeks to show that $\operatorname{Cl} (X \times \mathbb A^1) \simeq \operatorname{Cl} X$ for $X$ Noetherian, integral, separated, and regular in codimension 1. A point of codimension 1 in $X \times \mathbb A^1$ is…
Johnny Apple
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Existence of a morphism on given projective varitey (homework)

In need help in my homework assignment: let $X =\left \{ (x_{0}:x_{1}:x_{2}:x_{3}:x_{4})\in \mathbb{P}^{4} : rk\begin{pmatrix} x_{0} & x_{1} & x_{2}\\ x_{2} & x_{3} & x_{4} \end{pmatrix} < 2\right \} $ show that there is a morphism $\varphi…
GAJO
  • 155
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Bertini's theorem and hypersurfaces

I am reading "Algebraic Geometry, a first course", then I can't solve the following question that is an application of Bertini's theorem: Exercise $17.17$: Use Bertini's theorem to show that (a) the general hypersurface of degree $d$ in…
larsss
  • 81
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Inverse images under universally injective morphisms

Let Y be locally Noetherian, and consider a projective morphism $f:X \rightarrow Y$ such that the map $\textbf{Spec} f_\ast \mathcal{O}_X \rightarrow Y $ is universally injective. Let $C \rightarrow Y$ be a morphism of schemes with $C$ connected.…
Heidar Svan
  • 411
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Is this incidence variety in $\mathbb{P}^2 \times \mathbb{P}^2$ isomorphic to a variety in $\mathbb{P}^1 \times \mathbb{P}^1$?

I have an incidence variety $X = \{(p,\ell) \in C \times D^* : p \in \ell\} \subset \mathbb{P}^2 \times \mathbb{P}^2$, where $C = Z(f) \subset \mathbb{P}^2$ and $D^* = Z(g^*) \subset \mathbb{P}^2$ are two conics ($D^*$ is the dual conic of the conic…
Ricardo Buring
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Morphism $\Gamma :\mathbb{A}^n\rightarrow \mathbb{A}^d$ coresp. to $\gamma: k[y_1,\ldots,y_d]\rightarrow k[x_1,\ldots,x_n]$ is a projection

Let $X\subset \mathbb{A}^n$ be an affine variety with $A(X)=k[x_1,\ldots,x_n]/I(X)$. Write $A=k[a_1,\ldots,a_n]$ for $A(X)$ (where $a_i$ is the image of $x_i$). Let $B=k[y_1,\ldots,y_d]$ be a Noether normalization of $A$. I've previously shown that…
KJA
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Is Hartshorne exercise II.2.15(b) correct as written?

Here is the text of the exercise: If $f:X \rightarrow Y$ is a morphism of schemes over $k$, and $P \in X$ is a point with residue field $k$, then $f(P) \in Y$ also has residue field $k$. In the case where $X$ and $Y$ are spectra of fields, this…
Al-Biruni
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