Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

From where does the dualizing sheaf come from?

I have been going through basic Algebraic Geometry, and although I have been using Serre duality for quite a lot of time, I still do not understand how one came up with the dualizing sheaf for a Projective scheme $X \hookrightarrow \mathbb{P}^N.$…
Rio
  • 530
3
votes
1 answer

Algebraic Geometry problem with Nullstellensatz

I am right now battling with algebraic geometry and can't wrap my head around one problem. Problem: Assume that the field $K$ is the field of complex numbers. Let $V \in \mathbb{A}^n_K$ be an affine variety, let $f \in K[V]$. Suppose that for all $P…
Daria Dunina
  • 89
3
votes
2 answers

Stalk of the sheaf of regular functions on a subvariety

Suppose $Y$ is a subvariety of a variety $X$ (according to Hartshorne this means if $X$ is quasi-affine or quasi projective then $Y$ is a locally closed subset of $X$, c.f. exercise 3.10, chapter 1). Now given $i : Y \to X$ the inclusion map, I am…
user38268
3
votes
1 answer

Hartshorne Exercise II.6.1 Proving that $\operatorname{Cl}(X\times\mathbb{P}^n)\cong \operatorname{Cl}(X)\times \mathbb{Z}$.

I have been stuck on this computation for some time (at least a year now since I started learning AG seriously). The trick is to use Proposition II.6.5 and take the closed set $Z$ to be the hyperplane at $\infty$ to get a…
Shrugs
  • 1,458
3
votes
1 answer

Verifying Bertini's theorem for the hypersurface $V = \mathbb{V}(x^5 + y^5 + z^5 + t^5)$ in $\mathbb P^3$.

I want to verify the theorem of Bertini in the specific case of $V = \mathbb V(x^5 + y^5 + z^5 + t^5) \subset \mathbb P^3$. Because there are many different versions of this theorem, below the one I use. Let $V \subset \mathbb P^n$ be a smooth…
cattrainbow
  • 58
3
votes
1 answer

Irreducible plane curve of degree $d>2$ with a point $P$ of multiplicity $d-1$ is rational

We have to prove that an irreducible plane curve $X$ of degree $d>2$ with a point $P$ of multiplicity $d-1$ is rational and we need to find a resolution of this $X$. To prove that $X$ is rational, we can construct a birational map $\mathbb{P}^1…
emiel647
  • 81
3
votes
1 answer

Comparison of saturated ideals and radical ideals

Given a graded ring $B = A[x_0,\dots,x_n]$, $I$ a homogeneous ideal of $B$ not containing $B_+$. Then what are the relations between the racial ideal of $I$ and saturation of $I$? As far as I know, there are the following results, indicating…
Mathstudent
  • 351
3
votes
0 answers

Dimension of image of varieties

Set $k$ be an algebraically closed field. Let $X$ and $Y$ be two $k$-varieties.(one can assume they are projective) A question I have met many times is that: if $f$ is a morphism from $X$ to $Y$, then can we know $\operatorname{dim} f(X)\leq…
Richard
  • 1,402
3
votes
1 answer

Complex conjugation and homeomorphisms

According to this paper, Serre proved that there exists a pair $X, X'$ of smooth complex projective varieties, such that $X, X'$ are conjugate but not homeomorphic. Here, we say that $X, X'$ are conjugate if there exists a…
user142700
  • 2,840
3
votes
1 answer

Proving coefficients of a polynomial form a subvariety

I encountered this problem while studying algebraic geometry: Let $f \in \mathbb{C}[x, y, z]$ be a homogeneous polynomial of degree 3. The coefficients of $f$ represent a point $P_f$ in $\Bbb{P}^9$ . Show that $$\Bbb{P}^9\setminus\{P_f \,|\,…
abcX
  • 61
3
votes
1 answer

Identifying $\mathrm{Spec}(\mathbb C[x])$ with $\mathrm{Spec}(\mathbb C[x,y_1, y_2,\ldots]/(y_1^2, y_2^2,\ldots, (x-1)y_1, (x-2)y_2,\ldots))$

Vakil Rmk 5.2.2: Identify the following space with $\operatorname{Spec}(\mathbb C[x])$: $$X:=\operatorname{Spec}(\mathbb C[x,y_1, y_2,\ldots]/(y_1^2, y_2^2,\ldots, (x-1)y_1, (x-2)y_2,\ldots))$$ and then show that the nonreduced points of $X$ are…
Rodrigo
  • 7,646
3
votes
1 answer

Hartshorne Exercise II.6.9(a): Picard group of a singular curve

Exercise: *6.9. Singular Curves. Here we give another method of calculating the Picard group of a singular curve. Let $X$ be a projective curve over $k$, let $\tilde{X}$ be its normalization, and let $\pi: \tilde{X} \rightarrow X$ be the projection…
Richard
  • 1,402
3
votes
2 answers

Hartshorne Exercise III.2.7(a): sheaf cohomology of constant sheaf $\Bbb Z$ on $S^1$ in the usual topology

I am trying to solve this exercise: Let $S^1$ be the circle(with its usual topology), and let $\mathbb Z$ be the constant sheaf $\mathbb Z$ (a) Show that $H^1(S^1,\mathbb Z)\simeq \mathbb Z$, using our definition of cohomology. I have tried to…
Richard
  • 1,402
3
votes
0 answers

some question projection from a point

Let $p=[0,\cdots,0,1]\in \mathbb{P}^n$, $X \subseteq \mathbb{P}^n-\{p\}$ a projective variety and let $\overline{X}$ be the projection of from $p$ to $V(X_n)\cong \mathbb{P}^{n-1}$. I know that $\overline{X}$ is a projective variety using…
Sang Cheol Lee
  • 3,119
3
votes
2 answers

Questions about the local ring of a point on a variety.

Let $Y$ be a variety. Let $\mathcal{O}_{P, Y} = \mathcal{O}_{P}$ be the ring of germs of regular functions on $Y$ near $P$. That is, an element of $\mathcal{O}_P$ is pair $\langle U, f \rangle$ where $U$ is an open subset of $Y$ containing $P$ and…
LJR
  • 14,520
1 2 3
…
99 100