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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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Vector bundle with trivial chern classes

I saw in a paper that the only vector bundle of rank $r$ on the projective plane with trivial chern classes is the trivial vector bundle of rank $r$. I can see that the trivial vector bundle has trivial chern classes. How do we prove the converse?
user349424
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Direct image for standart affine cover of projective line

Let $U_1 \cong Spec(K[t])$ and $U_2\cong Spec(K[t])$ be a standard affine cover of a projective line $\mathbb{P^1}(K)$, where $K$ is some field. Let us denote open embedding as $j_k$ $$ j_k : U_k \to \mathbb{P^1}(K), k=1,2. $$ Is it correct that…
Alex
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Relate two descriptions of the stack $[X/G]$

I would like to understand the stack $[X/G]\to\mathfrak{Sch}$, where $X$ is a scheme and $G$ is an algebraic group acting on it. I found two descriptions of this groupoid, but I am unable to relate them, so any help in this direction would be…
Brenin
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Geometric Interpretation of Automorphisms of Projective Bundles

Let $\mathcal{E}$ be a rank three vector bundle on $\mathbb{P}^1$. It splits as $\mathcal{E}\cong\mathcal{O}(a_1)\oplus\mathcal{O}(a_2)\oplus\mathcal{O}(a_3)$. It's not hard to see that any automorphism of $\mathbb{P}(\mathcal{E})$ acts trivially on…
Samir Canning
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Topology in Arithmetic

Conjectured by Mordell and later proved by Faltings, a non-singular algebraic curve of genus $g$ over $\mathbb{Q}$ has finitely many rational points if $g > 1$. Since the genus of the Fermat curve $x^{n} + y^{n} = 1$ is $\frac{n(n-1)}{2}$ by the…
user02138
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Intersection with canonical divisor

Let $X\longrightarrow \mathbb P^1$ be a rational ellipic surface. In the Paper "pencils of cubic curves and rational elliptic surfaces" (by C.T.C.Wall) it is stated that C.K < 0, where C is a curve not contained in a fiber and K ist a canonical…
M. E.
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A question about Hartshorne Algebraic Geometry Chapter III Exercise 9.6

(I am not a native English speaker hence there may be some mistakes.) Recently I was working on the problem III.9.6 in Hartshorne, Algebraic Geometry. It states: Let $Y \subset \mathbb{P}^n$ be a nonsingular variety of dimension $\ge2$ over an…
ZetaW
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Topological Space on all Ideals

$\text{Spec}$ is a topological space on the prime ideals of a ring. What fails if we try to make the ideals into a topological space? We might try something like this: the points of this space are ideals. For $f_1, ..., f_n \in A$, put $D(f_1, ...,…
Ronald J. Zallman
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Picard Group of a Blowup

Let $\pi : \tilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowup of $\mathbb{P}^{3}$ along a smooth algebraic curve $C$ with exceptional divisor $E$ We know that $\text{Pic}(\mathbb{P}^{3}) \simeq \mathbb{Z}$. How do I determine the…
Allan Ramos
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direct image functor $f_*$ left exact

I would have to ask for apology for following question by everybody who is familar with algebraic geometry since this might be a quite trivial problem that cause some irritations for me: we consider a morphism $f: Z \to Y$ between schemes. then the…
user526728
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Question about plane quintic

Let the canonical curve $C$ $\subset$ $\mathbb{P}^5$ lie on the Veronese surface. How to see that $C$ is a smooth plane quintic?
Leo
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Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space $\mathbb{A}^3$ (over an algebraically closed field). Show that…
Math2012pc
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Show $I(V(f))=(f)$ when $f$ is irreducible.

This is a question from Perrin's text, and it goes like this: Let $k$ be algebraically closed. Let $F\in k[x,y]$ be an irreducible polynomial. Assume that $V(F)$ is infinite. Prove that $I(V(F))=(F)$. Here, $V(f)$ is the set of zeroes of the…
take008
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Hartshorne exercise I.2.14: question on Segre embedding

This is exercise I.2.14 in Hartshorne's Algebraic Geometry: Define $\psi : \mathbb{P}^{n}\times \mathbb{P}^{m}\longrightarrow \mathbb{P}^{N}$ where $N=rs+r+s$ by $(a_0,\dots,a_r)\times (b_0,\dots,b_s)=(\dots,a_ib_j,\dots)$ Show that $Im\psi$ is a…
Arsenaler
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Easy to state high-dimensional consequences of Bezout theorem

A classical consequence of Bezout's theorem for plane curves is Pascal's theorem. I am curious if there are some other statements that you find pretty that can be formulated (almost) as elementarily as Pascal's theorem and proven using higher…
agleaner
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