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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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finite flat subgroup schemes of smooth group schemes

Let $G$ be a smooth group scheme over $S$, and let $H\subset G$ be a finite flat closed subgroup scheme (hence it's a Cartier divisor). Let $nH$ be the closed subscheme corresponding to multiplying the Cartier divisor by the integer $n$. Is $nH$…
user355183
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Morphism of curve associated to an extension of fields of dim $1$

This is a basic question in algebraic geometry which probably is already answered somewhere in the web, but I am struggling too much on it and I couldn't find anything which could be of help.I thought that it is maybe worth to ask you if you can…
A. Prufrock
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Lift a variety over a number field to the complex numbers

Let's say I have a number field $k$ and a curve $C$ over $k$, can we lift $C$ to a variety $V$ over $\mathbb{C}$? It is like "complexifying" the curve. I read somewhere that it is possible to lift an elliptic curve $E$ over $k$ to an elliptic curve…
Natalie
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Equivalence of two definitions of proper morphism for separated varieties

In Hartshorne (and pretty much everywhere else I know) a proper morphism is defined to be A morphism of schemes $f:X\to Y$ is said to be proper if it is sperated, of finite type and universally closed. But then I started reading "Introduction…
Hamed
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Why is $\mathbb{C}^2\setminus\{(0,0)\}$ not a basic open set?

Consider the affine variety $\mathbb{C}^2$ equipped with Zariski topology. By the question above, I mean why $X:=\mathbb{C}^2\setminus\{(0,0)\}$ cannot be written as $$ X:=U_f:= \{(x,y)\in \mathbb{C}^2:f(x,y)\neq 0\} \mbox{ for some…
user166467
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Morphism of algebraic varieties with vanishing differential

In differential geometry, we know that given a smooth map between smooth manifolds $\phi:X\to Y$, such that $X$ is connected and $d\phi|_x\equiv0$ for all $x\in X$, then $\phi$ is constant (this follows directly from the fact that a real function on…
KotelKanim
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Projection from a point on a curve

Let $k$ be an algebraically closed field, char $k=0$, and let $C\subset\mathbb{P}_k^2$ be a nonsingular projective plane curve of degree $d$. Let $O\in C$, $L\subset\mathbb{P}_k^2$ a line not containing $O$, and consider the map $\varphi:C\to L$…
freeRmodule
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Example $3.3.1$ in Hartshorne

Let $k$ be an algebraically closed field, and let $$X = \operatorname{Spec} k[x,y,t]/(ty-x^2)$$ $$Y = \operatorname{Spec} k[t]$$ Hartshorne comments that both schemes $X$ and $Y$ are of finite type over $k$. This means that the natural morphisms $X…
user7090
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Definition of degree of a coherent sheaf

Let $E$ be a coherent sheaf on a scheme $X$. Let $d = \text{dim}X$ be the dimension of $E$. Huybrechts and Lehn define the degree of $E$ to be: $$ \text{deg} E := \alpha_{d-1}(E) - \text{rk}(E)\cdot\alpha_{d-1}(\mathcal{O}_X) $$ where $\alpha_i$ is…
baltazar
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Do I not understand modern geometrical objects, or is it because there are none?

I'm a physicist and so never had a lecture on algebraic geometry. What I really try to find out for some time now (and it's funny how I don't seem to be able to find it out) is if the modern theory, since the 60's say, is concerned with new objects…
Nikolaj-K
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Chern classes of the associated vector bundle of a branched covering

Let $f \colon X \to \mathcal{Q}_7$ be a branched covering of degree $3$ of a $7$-dimensional smooth projective quadric $\mathcal{Q}_7 \subset \mathbb{P}_8$, where $X$ is a smooth connected projective variety. We can define the associated vector…
L_K666
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Comparing morphisms of local nature for equivalent topologies

Let $S$ be a scheme. It is known that the smooth topology on $\textrm{Sch}_S$ is equivalent to the étale topology, basically because every smooth covering can be refined to an étale covering. Question. Is it true, then, that all properties P of…
Brenin
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Difficulty calculating with rational functions

I'm having trouble calculating the image of a point $P=[0:0:1]$ under a polynomial map:$$f:V \to \mathbb{P}^1 : [x:y:z] \mapsto [y:x]$$ where $$V = \left\{ [x:y:z] \in \mathbb{P}^2 ~\mid~ zy^2 = zx^2 + x^3 \right\}.$$ What is $f([0:0:1])$? Dead…
Jack Schmidt
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Exercise 1.11 Harris Algebraic Geometry: A First Course

I am trying to do part (b) of Exercise 1.11 in Harris' book Algebraic Geometry: A First Course. Let $F_0=Z_0Z_2−Z_1^2$, $F_1=Z_0Z_3−Z_1Z_2$, $F_2=Z_1Z_3−Z_2^2$ (s.t. $V(F_0,F_1,F_2)$ is the twisted cubic). Define $F_λ=λ_0F_0+λ_1F_1+λ_2F_2$. Prove…
Justine
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Function field of the projective line

Suppose I choose two rational functions, say, $$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5 (4+t)}{(1+4t)}.$$ Then I know that $K(X):= \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y):=\mathbf{C}(t)$, then…
Philoi
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