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
5
votes
1 answer

some question of sheaf

Let $X$ be a topological space, $F$ an abelian sheaf on $X$ and $Z$ be a locally closed subset. Then, we can choose an open subset $V$ such that $Z \subset V$ and $Z$ is a closed subset in $V$. Define $\Gamma_Z(X,F)$ the subgroup of $F(V)$…
Sang Cheol Lee
  • 3,119
5
votes
1 answer

Why is it called the 'Seesaw theorem'?

I'm reading Mumford's "Abelian variety" and he proved the theorem of cubes by using the Seesaw theorem: Let $X$ be a complete variety, $T$ any variety and $\mathcal{L}$ a line bundle on $X\times T$. Then the set $$ T_{1} = \{t\in T\,:\,…
Seewoo Lee
  • 15,137
  • 2
  • 19
  • 49
5
votes
1 answer

Frobenius morphism and global sections of direct image of structure sheaf

Let $X$ be a proper scheme defined over an algebraically closed field of characteristic $p > 0$. Let $F : X\rightarrow X$ be the absolute Frobenius morphism. What is the dimension of $H^0(X, F_*\mathcal{O}_X)$?
user8788
5
votes
1 answer

Genus of a complex curve of degree $2n$

One basic alg geom book convinced me that the genus of a complex curve given by $y^2=x^{2n}+a_{2n-1}x^{2n-1}+...+a_0$ is $n-1$. Another textbook computes the genus of a singular variety given as $(y-b_1x)(y-b_2x)\cdot\cdot\cdot(y-b_{2n}x)$ to be…
Michael
  • 1,649
  • 10
  • 16
5
votes
2 answers

Homomorphism of coordinate rings induces a polynomial map

I have a question about the proof of Proposition 2 on page 26 of Fulton's algebraic curves. Let $V\subset \mathbb A^n$ and $W\subset \mathbb A^m$ be affine varieties, and let $\Gamma(V)$ and $\Gamma(W)$ be their coordinate rings. Suppose that we…
Shimrod
  • 519
5
votes
2 answers

How to define the dual sheaf

If $X$ is a scheme and a sheaf $\mathcal F$of modules on $X$, then how can we define the dual $\mathcal F^*$ of $\mathcal F$? Obviously, we set $\mathcal F^*(U)=\rm{Hom}_{\mathcal O_X(U)}(\mathcal F(U),\mathcal O_X(U))$ and use sheafication, but how…
Summer
  • 6,893
5
votes
1 answer

pushforward and pullback sheaf is isomorphism?

I am reading Hartshorne's book chapter 5 (on surface) and I have a question: on page 371, proposition 2.3, it says: Let $X$ be surface, $C$ curve, $\pi:X\to C$ ruled surface. $f$ be a fiber, $\sigma$ be a section of $\pi$, $C_0=\sigma(C)$. Let $D$…
User X
  • 1,081
5
votes
1 answer

Why is this morphism of vector bundles given by a matrix of linear forms

Let $X$ be a smooth hypersurface in Projective space $\mathbb{P}^n$ of degree $ d$ defined by the equation $f=0$. Given that we have a vector bundle $E$ of rank $r\geq1$ on $X$ such that we have the following exact sequence on…
gradstudent
  • 3,372
5
votes
0 answers

Euler characteristic of a variety and its analytification

Let $X$ be a smooth projective complex variety and $\mathcal{F}$ a coherent sheaf on $X$. Let $\tau$ be a Grothendieck topology and $$ \chi(X,\mathcal{F},\tau)=\sum_i(-1)^i dim_{\mathbb{C}}H^i_\tau(X,\mathcal{F}) $$ the Euler characteristic of…
user58341
  • 51
5
votes
0 answers

Relative Fano variety of lines isomorphic to the usual one?

Let $\mathfrak X\to \mathbb P^1$ be a projective family, where $\mathfrak X\subset \mathbb P^n\times\mathbb P^1$. Then we have the relative Fano variety of lines $F(\mathfrak X/\mathbb P^1)$; besides, using Segre embedding, $\mathfrak X\subset…
Akatsuki
  • 3,250
5
votes
0 answers

Helpful examples in learning Abhyankar's conjecture for the affine line?

I'm reading Raynaud's proof for the Abhyankar conjecture on $\mathbb{A}^1$, namely, Let $k$ be an algebraically closed field of characteristic $p>0$. Then every quasi-$p$ group is the Galois group of some connected etale cover of $\mathbb…
user31480
  • 479
5
votes
0 answers

Is the relative rank function with respect to an ample line bundle non-decreasing

Let me make the question in the title more precise. Let $f:X\to $ Spec $k$ be a smooth projective connected variety over a field $k$ of characteristic zero. Let $\mathcal L$ be a line bundle on $X$. In my set-up this line bundle has many positivity…
Harry
  • 3,063
5
votes
1 answer

An analogue of degree-genus formula for surfaces.

I have recently learnt Riemann-Roch formula for surfaces. Roughly speaking, the theorem says that on a reasonably nice surface we have the relation: $$ \chi(D) = \frac{1}{2}(D.D - D.K) + p_a + 1 $$ where $D$ is a divisor, $\chi$ is the Euler…
Jakub Konieczny
  • 12,588
5
votes
1 answer

Question on Veronese map

On page 10 of this note the author proved that the image of $\mathbb{P}^{n}$ under the Veronese map $\nu_{d}$ : $\mathbb{P}^{n}\longrightarrow \mathbb{P}^{m}$ is isomprphic to $\mathbb{P}^{n}$. In the proof, the author construct an inverse map :…
knot
  • 671
5
votes
1 answer

Equivalence of ample divisor

Let $X$ be a smooth complex projective variety. Having ample anticanonical class is equivalent to having $-K_{X} > 0$? How to prove that? Does it hold for any ample divisor in $X$?
rla
  • 2,345
1 2 3
…
99 100