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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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The graph of a rational map

Let $X$ and $Y$ be smooth projective varieties and let $f:X - \to Y$ be a rational map defined on $U$. Define the graph of this rational map to be the closure of $\{(u,f(u)): u\in U\}$ in $X\times Y$. Is the graph necessarily smooth?
user64480
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Finding a hypersurface whose intersection of Veronese surface is the given curve?

I'm working on Hartshorne's Algebraic geometry exercise I.2.12. It says: Let $Y$ be the image of the 2-uple embedding $\rho:\mathbb{P}^2\rightarrow\mathbb{P}^5$. If $C\subset Y$ is a closed curve (a curve is a variety of dimension 1), show that…
noobgi
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Differences Between Studying Scheme Theory with and Without Cohomology

I am curious about the main ways the geometric intuition is different in studying schemes with cohomology (ie Hartshorne) vs. studying schemes without cohomology (ie Eisenbud-Harris). It would be very useful to know how the types of geometric…
HinLear
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Hartshorne Exercise II 3.22 (a) - Dimensions of the Fibres of a morphism

I'm currently stuck at Exercise II 3.22 (a) in Hartshorne's Algebraic geometry, which states Let $f: X \to Y$ be a dominant morphism of integral schemes of finite type over a field $k$, and let $Y' \subset Y$ be an irreducible closed subset whose…
red_trumpet
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Existence of ample divisor with smooth image

Let $f:X \rightarrow Y$ be a proper, generically finite morphism between smooth, projective varieties. Is there an ample divisor $D$ on $X$ such that $f(D)$ is smooth?
safety stegosaurus
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Harris' AG ex 2.24: projective variety under regular map.

I am trying to show that if $X$ is an irreducible projective variety, its graph under a regular map into projective space is an irreducible subvariety [Harris Ex 2.24]. I tried to modify a proof I gave before for the affine case. If $X\subset P^n$…
Steven-Owen
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Definition of Regular function

In Harsthone page 15, the notion of regular function of a quasi affine variety $Y$ is defined as followds. A function $f:Y \to k$ is regular at a point $P \in Y$ if there is an open neighborhood $U$ with $p \in U \subset Y$ and polynomial $g,h…
Jean Billie
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Intersection between a curve of degree six and a conic

I've seen this exercise in the book "Lectures on curves, surfaces and projective varieties" by Beltrametti and others: Show that three polynomials $\phi_2(x_1,x_2,x_3), \phi_3(x_1,x_2,x_3), \phi_4(x_1,x_2,x_3)$ homogeneous of degree $deg(\phi_i)=i$…
Cirdan
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Intersection of irreducible sets in $\mathbb A_{\mathbb C}^3$ is not irreducible

I am looking for a counterexample in order to answer to the following: Is the intersection of two closed irreducible sets in $\mathbb A_{\mathbb C}^3$ still irreducible? The topology on $\mathbb A_{\mathbb C}^3$ is clearly the Zariski one; by…
Romeo
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Number of minimal sections in a (geometrically) ruled surface

Let $\mathbb{P}(X)$ be a non-trivial $\mathbb{P}^1$-bundle over a curve $C$ (here $X$ is a vector bundle of rank 2 over $C$). A minimal section of $\mathbb{P}(X)$ is a section of minimal self-intersection. My questions are the following: Can…
Jimmy
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Is the ring of global functions on an integral scheme integral?

Is the ring of global functions on an integral scheme an integral domain (if the scheme is affine then it is try by definition so we are interested in non-affine schemes)? It is necessarily a reduced ring (https://stacks.math.columbia.edu/tag/01OL)…
user691994
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Question about proof of Hartshorne book Lemma III.2.4

Let $(X,\mathcal{O}_X)$ be a ringed space. For any open subet $U \subseteq X$, Let $\mathcal{O}_U$ denote the sheaf $j_!(\mathcal{O}_X|_U)$ which is the restriction of $\mathcal{O}_X$ to $U$, extended by zero outside $U$. Let $\mathcal{I}$ be an…
Sang Cheol Lee
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Must two morphisms into an algebraic space which agree on closed points be the same?

First I apologize if this is elementary. I have just started looking at the basics of stacks and algebraic spaces so my understanding is lacking. Let's work over an algebraically closed field $k$. Suppose I have an algebraic space $\mathcal{A}$ and…
user16544
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some question of $\mathcal{O}_X$-module

Let $X$ and $Y$ be schemes and let $F$ be a sheaf on $Y$. Let $f: X \rightarrow Y$ be a morphisme of schemes. Define the inverse image of $F$, $$f^*F:= f^{-1}F\otimes_{f^{-1}\mathcal{O}Y}\mathcal{O}_X$$ For an open subset $U\subseteq X$, $f$ be the…
Sang Cheol Lee
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Finite bijective morphism to variety with separable function field

Let $V$ be an $n$-dimensional variety over $k$. The function field $k(V)$ doesn't have to be separable over $k$ but I'd like to know which conditions imply that we can find a finite bijective morphism from $V$ to a variety $W$ with separable…
MichalisN
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