Mathematics Stack Exchange
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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.

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Morphism of projective schemes not induced by maps of graded rings, Vakil's 16.4 G

Define the graded rings $R_\bullet =k[u,v,w]/(uw-v^2)$ and $S_\bullet=k[x,y]$, where $u,v,w,x,y$ all have degree $1$. Then it is obvious that $\text{Proj}~R_\bullet$ and $\text{Proj}~S_\bullet$ are both $\mathbb{P}^1_k$, while there is an…
Wenzhe
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Counterexample Question

Let $f:X\rightarrow Y$ be a morphism of varieties. If $f(X)$ is dense in $Y$, then $\tilde{f}:\Gamma(Y)\rightarrow \Gamma(X)$ is injective, where $\tilde{f}$ is the homomorphism induced by $f$. In fact, if $X$ and $Y$ are affine, then we have if…
John S
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When does a morphism preserve the degree of curves?

Suppose $X \subset \mathbb{P}_k^n$ is a smooth, projective curve over an algebraically closed field $k$ of degree $d$ . In this case, degree of $X$ is defined as the leading coefficient of $P_X$, where $P_X$ is the Hilbert polynomial of $X$. I…
Li Zhan
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Does blow up of subscheme in special fiber change the generic fiber?

Let $X\to \mathrm{Spec}(R)$ be a finite type scheme over DVR, choose a closed subscheme $Y$ of the closed fiber $X_0$ and blow up $Y$ in $X$, will the generic fiber always remain the same?
user93417
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If $X$ has non-singular normalization $\dim (\mathrm{Sing(X)})=\dim (X)-1$?

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety, and let $$ \nu:X^{\nu}\rightarrow X $$ be its normalization. Let us suppose that $X^{\nu}(\neq X)$ is smooth. I wonder if in this case $$ \dim (\mathrm{Sing(X)})=\dim (X)-1. $$ I think Serre's…
Carlito
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codimension of "jumping" of the dimension of fibers

Let $f:X\rightarrow Y$ be a dominant morphism of projective (and smooth if you like) varieties over an algebraically closed field $k$ such that $n=\dim(X)=\dim(Y)$. Then $f$ is proper, so by Chevalley's upper semi-continuity theorem, $\dim(X_y)$ is…
user16544
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Exact sequence in Beauville's "Complex Algebraic Surfaces"

On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent post, I was explained that you can take $s$ and…
rfauffar
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Every variety is isomorphic to an intersection of a linear space and a Veronese surface

"Deduce that any projective variety is isomorphic to an intersection of a Veronese variety with a linear space" I've been trying to solve this exercise from Joe Harris book. I can see that if a variety $X\subset \mathbb{P}^n$ has only polynomials of…
ett
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If a principal divisor is defined over K, then is the function?

Let $X$ be an algebraic variety, $D$ a principal divisor of $X$ defined over $K$, i.e. the points of $D$ are in $X(K)$ and there is a function in $\overline{K}(X)$ whose divisor is $D$. Is $D$ necessarily the divisor of a function on $X$ defined…
Tony
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For a Noetherian scheme X, show that $X_{red}$ affine implies $X$ affine.

I have the following problem: Let X be a Noetherian Scheme and suppose that $X_{red}$ is affine. Show that this implies that X is affine. OK, so I know the "classical" proof of this using Serre's criterion for affineness and with cohomology.…
Dedalus
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Smoothing double points on Barth's decic surface

Here is the equation I have: $8(x^2-\phi^4y^2)(y^2-\phi^4z^2)(z^2-\phi^4x^2)(x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2)+(3+5\phi)(x^2+y^2+z^2-1-c)^2(x^2+y^2+z^2-\phi^2-b)^2-a=0$ Where $\phi = \frac{1+\sqrt 5}{2}$. I'm trying to form 3D-printable models of…
den-chan
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Is every one-to-one morphism between varieties necessarily a homeomorphism?

Let $f$ be a morphism between two irreducible varieties, and one-to-one. Is $f$ actually a homeomorphism onto its image? Here the varieties are equipped with Zariski topology. I know if the varieties are projective then it is true. (Because…
Akatsuki
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How can we interpret derivations as elements of the tangent sheaf

Suppose $X$ is an algebraic variety and $\delta : X \to X \times X$ is the diagonal map. I am defining the cotangent sheaf $\Omega^1_X$ as $\delta^{-1}(I/I^2)$ where $I$ is the ideal sheaf of functions in $\mathcal{O}_{X\times X}$ which vanishes on…
Mathmo1123
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Birational morphism

I have a question on rational and birational maps: Is the map $$\mathbb{P}^1\rightarrow \mathbb{P}^2, (x:y) \mapsto (x:y:1)$$ rational? Birational? If birational what is its inverse? Same questions for map $$\mathbb{P}^1 \rightarrow \mathbb{P}^2,…
user17090
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regular functions: two definitions

Let $X$ be an (affine) algebraic set i.e. the zeros' locus of a set of polynomial $S\subseteq k[X_1,\ldots,X_n],$ Let's look at these two definitions: 1) A regular function in $p\in X$, is an element of the following ring: $$\mathcal…
Dubious
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