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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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stereographic projection in a projective curve

It seems that there is a standard technique for an idea similar to the stereographic projection. I don't know how can I use it. For example here in this exercise, how can I use it ? I'm really sorry for being so stupid.... Define a birational map…
Matias
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Is there a proof of Noether's formula without using general Riemann-Roch theorem?

Is there an algebraic proof of Noether's formula $\chi(O_S)=\frac{1}{12}(K^2+c_2(T_S))$ without directly applying general Riemann-Roch theorem?
user93417
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Boundedness of a set

Let $$S = \{(x, y) \in \mathbb{R}^2: x^2 + 2hxy + y^2 =1\}$$ For what values of $h$ is the set $S$ nonempty and bounded? For $h = 0,$ it is surely bounded, the curve being the unit circle. What for other $h$? Please help someone.
user242028
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tangent space at some point of a quasi-projective variety

In order to define the tangent space to a quasi-projective variety $V$ (i.e a locally closed closed subset of $\mathbb{P}^n$ considered with Zariski topology induced from $\mathbb{P}^n$) at a point $p$ we must think of $V$ as an open subset in a…
Myshkin
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How small can affine sets be?

Quick question: let $V$ be a closed subset in the Zariski topology on $k^n$ ($k$ algebraically closed), and let $U$ be an open subset of $V$. Is it possible to find another open subset $U_1$ of $V$ such that $U_1$ is contained in $U$, and the…
D_S
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Closure of image of algebraic variety

Suppose $A$ is algebraic variety and $f$ is a regular map from $A$ to grassmannian. Let us consider closure of $f(A)$ in Zariski and analitical topology. Is it true that these closures coincide?
Alex-omsk
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Showing every finite rank n vector bundle over the affine line is trivial of rank n

In Ravi Vakil's algebraic geometry notes, exercise 13.2.C asks to show that every finite rank n vector bundle over $X = \mathbb{A}^1_k$ is actually free of rank n. The hint is to use the structure theorem for finitely generated modules over PIDs. My…
Garnet
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"Every regular function on an affine variety is polynomial" - generalisation to the case of a reducible variety

$K$ is an algebraically-closed field, and for an affine variety $X$, $A(X) $ denotes the ring of polynomial functions on $X$. What I would like to prove is the following: "Let $X \subset \Bbb{A}^n(K) $ be Zariski-closed, and $ f: X \rightarrow K …
Latimer Leviosa
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Normal bundle of zero scheme of section

Suppose $Y$ is a smooth variety, $E$ is a rank $d$ bundle on $Y$, $s$ is a regular section of $E$ over $Y$,(i.e.,locally under a trivialization $E|_U\cong O_U^d$, write $s=(s_1,\dots,s_d)$, then $s_i$ form a regular sequence). Suppose $X$ is the…
user93417
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1 answer

Number of birational classes of dimension d, geometric genus 0 varieties?

Fix an algebraically closed field $k$ and a positive integer $d$. My question is, what is the number of birational classes of dimension $d$, projective varieties over $k$ with geometric genus 0? If it makes the question answerable over fields where…
user16544
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Is this surface rational -- Magma says no, but I have rational parametrization?

Let $f(x,y,z)$ be the degree $6$ polynomial: x^6 + 6*x^5 + 15*x^4 - 3*x^2*y^2 + 20*x^3 - 18*x^2*y - 6*x*y^2 - 2*y^3 - 12*x^2 - 36*x*y - 21*y^2 + 4*z^2 - 48*x - 72*y - 80 I am interested if the surface $f(x,y,z)=0$ is rational. We have…
joro
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Degree of a projective variety

Let $X \subset \mathbb{P}^n$ be a projective variety of dimension $k
Manos
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Definitions of $\mathcal{O}(n)$ and sheaves associated to a module

In Vakil's Foundations of Algebraic Geometry (see p. 385) he defines the sheaf $\mathcal{O}(m)$ on $\mathbb{P}^n$ to be equal to $\mathcal{O}$ on affine sets $U_i = $Spec $k[x_{0/i}, x_{1/i}, \dots, x_{n/i}]$ (his notation; this is the open subset…
Kopper
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the variety of an irreducible polynomial sum of two homogeneous polynomials of degree m, m+1 is rational

This problem has two parts, the first I did it. But I have problems with the second. Given an irreducible polynomial F over $k[T]$. Prove that the ring $ k\left[ {T_1 ,...,T_n } \right]/\left( {I\left( {V\left( F \right)} \right)} \right) $, is an…
Matias
  • 207
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Fiber dimension theorem for locally closed sets

I want to prove (or to find a reference to) the following statement: Statement: Let $Z$ be an irreducible locally closed set (Zariski topology) of $\mathbb C^n$ and $\pi$ be a projection on the first $l$ coordinates (the values of $l$ is not…
guest_09072015
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