Mathematics Stack Exchange
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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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Picard group of generic fibre

Let $C$ be an irreducible curve over a field $k$ and let $X$ be a $k$-variety equipped with a morphism $f: X \to C$. Let $X_{k(C)} \to k(C)$ be the generic fibre of this morphism. Under which "reasonable" conditions on $X$, $C$ and/or $f$…
Evariste
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Fibres of the base change of a scheme

I am trying to gain a better understanding of the notion of fibre products of schemes. Two major applications that I've began to study are: 1) Making an $S$-scheme $X$ into an $S'$-scheme via a morphism $S' \rightarrow S$, and 2) Obtaining the…
Paul
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Why is $\operatorname{Spec} \ k$ final in category of $k$ schemes?

I am working on an exercise trying to show that $Spec \ k$ is final in category of $k$ schemes. I am stuck and I would appreciate any assistance. Thank you! PS The definition I have for $k$ scheme is that it is a morphism of the form $X \rightarrow…
user211392
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Irreducible conics

An algebraic set (not necessairly a variety) $X \subseteq \mathbb{A}^2$ defined by a polynomial of degree $2$ is called a conic. The problem is: Show that any irreducible conic is isomorphic either to $Z(y-x^2)$ or to $Z(xy-1)$ after an affine…
Leafar
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Riemann-Roch theorem without heavy tools

I have read two proofs of Riemann-Roch : one very quick in Forster, Lecture on Riemann Surfaces which use cohomology of sheaf, and results from functional analysis. Another one is in the book of Miranda about Riemann surfaces, which is more…
user171326
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Generically finite morphism of surfaces

Let $S_1$ and $S_2$ be smooth projective surfaces over $\mathbb{C}$ and let $f: S_1 \to S_2$ be a morphism which is generically finite of degree $d$. How does one prove that $f_* f^* D = dD$ for all divisors $D$ on $S_2$? This shouldn't be hard, but…
Evariste
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Why is the domain of a rational function necessarily nonempty

Let $V$ be an irreducible affine variety. A rational map $f : V \to \mathbb A^n$ is an $n$-tuple of maps $(f_1, \ldots , f_n)$ where there $f_i$ are rational functions i.e. are in $k(V)$. Th map is called regular at the point $P$ if all the $f_i$…
Matt
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Unions and intersections of algebraic varieties

Let A = $k[x_1, x_2, \ldots, x_n]$ and let $I_{\lambda}$ be an ideal of A. Let J = $\sum_{\lambda \in \Lambda} I_{\lambda}$ be a finite sum. Show that $V(J) = \cap_{\lambda \in \Lambda} V(I_{\lambda})$. It seems quite obvious that $V(\sum…
Mary
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question concerning a proof in Liu's "Algebraic Geometry and Arithmetic Curves"

I'm following the proof of Prop. 4.4., Chapter 7.4. Liu-Algebraic Geometry and Arithmetic Curves: $\textbf{Proposition 4.4.}$ Let $X$ be a smooth, geometrically connected, projective curve over a field $k$ of genus $g$. Let $\mathcal{L}$ be an…
user220164
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The regular functions on an open subset $U$ of an affine variety $X$ are the polynomials on $A(X)$.

This is an exercise of Gathmann's Algebraic Geometry. Show that the regular functions on an open subset $U$ of an affine variety $X$ are the polynomials on $A(X)$, if $A(X)$ is a UFD and $U$ is the complement of an irreducible subvariety of…
KittyL
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Different definitions of Kodaira dimension

Let X be a smooth projective variety with canonical class K. Let a be defined to be the maximum dimension of the image of X under the rational map induced by the linear system |nK| as n ranges over all positive integers. Let b be defined to be the…
Robert Garbary
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Two linear subspaces are isomorphic as algebraic sets if and only if they have the same dimensions

This is an example from Karen Smith's notes. Let $V_1, V_2 \subset k^n$ be linear subspaces (defined by some collection of linear polynomials). Then $V_1 \cong V_2$ as algebraic sets if and only if $\dim(V_1)=\dim(V_2)$. I am not sure how to start.…
KittyL
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What is the difference between Hom and Sheaf Hom?

I'm reading Hartshorne's book, and in 3.6 he begins to go into detail about Ext and sheaf Ext, which are derived functors of Hom and sheaf Hom respectively. Let $\mathcal{F,G}$ be sheaves of $\mathcal{O}_X$ modules on a scheme $X.$…
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Torsion-freeness is not affine local

I am working on an "unimportant" exercise (c.f. Vakil, exercise 13.5.J) which goes as follows Exercise 13.5.J: Find an example on a two-point space showing that $M:=A$ might not be a torsion-free $A$-module even though…
enoughsaid05
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What does the spectrum of the Grothendieck ring of varieties look like?

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation $[Y]=[X]+[Y\setminus X]$ whenever $X$ is a closed subvariety of $Y$.…
Brenin
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