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
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Ideal of the fiber of the thickening of a point

Let $\phi : A \to B$ be a morphism of rings and let $f : X \to S$ be the corresponding morphism of schemes. let $\mathfrak{q}$ be a prime ideal of $A$ corresponding to a point $s$ of $S$. Let $T = Spec(\mathscr{O}_{S,s}/\mathfrak{m}_s^n)$ be a…
DonDon
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Equality of stalks implies equality on a neighborhood

Let $X$ be a scheme, $Z$ be a closed subscheme of $X$ and $z$ be a point of $Z$. In the following notes : http://web.mit.edu/~holden1/www/coursework/math/18.787/main.pdf, page 43, claim 6.3 I believe that it is said that to show that $z$ is…
ChetB
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Symmetric product of global sections on a curve

Let $C$ be a complete curve defined over complex number, $D$ a effective divisor on $C$, and $V\subseteq H^0(D)$ a subspace. Assume that $V$ is base-point-free, which means that the zero of the sections in $V$ do not have common point. Consider the…
Pyramid
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Fibrations in algebraic geometry

Suppose that $f:X\rightarrow Y$ is a morphism of finite type schemes over an algebraically closed field $k$. Assume that for every closed point $y\in Y$ the fiber $X_y$ of $f$ in is isomorphic to $\mathbb{P}^n_k$. Now let $y_0\in Y$ be…
asker
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Locally closed immersions are locally of finite type

I am writing the proof that locally closed immersions are of finite type but I am stuck at minor detail. I would like that either (1) preimages of open affines by open immersions be quasicompact or (2) that given an affine $Spec\, A$ in the source…
Rodrigo
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base change of a reduced scheme over $\mathbb{Z}$

I'm reading through the stacks project and came across a lemma along these lines: Let X be a scheme over a perfect field k. Then, $X$ is reduced implies $X$ is geometrically reduced. here is my question: does this lemma extend to schemes over rings…
adrido
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How can I analytically determine a 1D projection of a high-dimensional ellipsoid?

For an axis-aligned ellipsoid $a_{11}x_1^2+a_{22}x_2^2+a_{33}x_3^2+a_{44}x_4^2=1$ it is easy to see that $x_1$ is in the range $-1/\sqrt{a_{11}}$ to $1/\sqrt{a_{11}}$. However, in the general case of a rotated ellipsoid, $\sum_{i,j} a_{ij}x_ix_j=1$…
Jonas Heidelberg
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Proposition 4.3.9 in Liu: flatness by domination

Proposition 4.3.9 in Liu says: Let $Y$ be a Dedeking scheme. Let $f:X\to Y$ be a morphism with $X$ reduced. Then $f$ is flat if and only if every irreducible component of $X$ dominates $Y$. I don't understand details in the proof: the proof is going…
Gabriel Soranzo
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Computing an Euler characteristic

Let $X$ be an abelian variety of dimension $3$. I wish to compute the Euler characteristic of a certain subscheme $V$ of the Hilbert scheme $H=\textrm{Hilb}^3X$. Definition of $V\subset H$: it consists of those $[Z]\in H$ supported on two distinct…
Brenin
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Does the image of the projection morphism of a fiber product contain an open set?

Let $f: X\times_S Y \to X$ be the projection morphism in the definition of fiber product.$ U\subset X\times_S Y$ be an open set. Does $f(U)$ contain a non-empty open set of $X$? I know this can be reduced to the affine case. If it is not true in…
Li Zhan
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How do you get an algebraic equation from a parametrization?

I know the Cycloid is an example of a curve with a parametrization that doesn't have an algebraic equation. But is there a general procedure to find the algebraic equation of parametric curves that do have one? (I am mostly concerned with plane…
Rodrigo
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The Uniformization Theorem and Elliptic Curves

In the theory of elliptic curves, I have read that the elliptic curves is topologically equivalent to a torus, given by $\mathbb{C}$/ $\Lambda$, where $\Lambda$ is a lattice. The proof appears to use the Uniformization Theorem, which states that…
math1234567
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Codimension of a variety

I am reading the book algebraic geometry: a first course by Joe Harris. I have a question about codimension of a variety. Usually, if we have a subspace $W$ of dim $k$ in a space $V$ of dim $n$, then $W$ is of codimension $n-k$. But I cannot…
LJR
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Questions about prime divisors.

Let $Y$ be a prime divisor on $X$ and $\eta \in Y$ its generic point. Then the local ring $\mathcal{O}_{\eta, X}$ is a discrete valuation ring with quotient field $K$, the function field of $X$. This is in the last paragraph on page 130 of…
LJR
  • 14,520
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Isomorphism of function field

Let $X$ be a prevariety. The function field $k(X)$ of $X$ is defined to be the inverse limit of $O_{X}(U)$, where $U$ is non-empty open set of $X$. Question:how to give the explicit isomorphism between $k(X)$ and $k(V)$, where $V$ is any open…
yang
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