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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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Unique group law on cubic

I'm looking at the following problem from Pg. 47 of Miles Reid's Undergraduate Algebraic Geometry (https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf): 2.11 (Group law on cuspidal cubic.) Consider the curve $$C:(z=x^3)\subset…
ArbitraryElement
  • 347
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A generalization of the concept of affine variety

I'm studying Fulton's book and in the chapter $6$, he gives a definition of variety. Before that he defines the affine variety in the chapter 2: The definition of affine variety is The definition of variety: This definition of variety is a…
user42912
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Dimension of complete linear system $|D|$, where $D=aE$ with $|E|$ base point free pencil

Let $X$ be a complex projective surface, and let $D \in \mathrm{Div}(X)$. Assume that $D=aE$ for some $a \in \mathbb N$, where $E \in \mathrm{Div}(X)$ is a divisor such that $|E|$ is a pencil (i.e. $h^0(E)=2$) with empty base locus. Is it true or…
Francesco Genovese
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Criteria for an Immersion into Projective Space

I am doing Exercise II.7.7c in Hartshorne (not homework). In this problem, he asks to show that the linear system of conics in $\mathbb{P}^2$ passing through a fixed point $P$ gives an immersion of $\mathbb{P}^2-P$ into $\mathbb{P}^4$ (over $k$…
Cass
  • 2,987
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Why does $y^2=(x^2-1)(x^2-4)(x^2-9)$ have genus 2? Why not genus 10? Isn't it nonsingular?

If a curve is nonsingular then the arithmetic genus and the geometric genus should be the same. Using the genus-degree formula, I obtain that the genus of this curve should be 10. However, the book I'm reading states that this curve has genus 2.
Simon M
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Are the nonmaximal point in $\operatorname{Spec} A$ unimportant in some situations?

(I don't know the best way to set up the context of this question, so I am just going to say whatever I heard people keep saying, according to my poor memory.) Let $k$ be an algebraically closed field. An affine $k$-variety is the space…
Ray
  • 1,280
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Jacobian computes the Zariski cotangent space at a rational point

In Vakil's Foundations of Algebraic Geometry (July 31, 2023 version), he has the following Exercise (13.1.I): Suppose $X$ is a finite type $k$-scheme. Then locally it is of the form $\operatorname{Spec} k[x_1, \dots , x_n]/(f_1, \dots , f_r)$. Show…
mattematician
  • 192
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cohomology of projective space

Hartshorne book (III.5.1) say that the natural map $$H^0(\mathbb{P}^r_{A},\mathcal{O}(n)) \times H^r(\mathbb{P}^r_{A},\mathcal{O}(-n-r-1)) \rightarrow H^r(\mathbb{P}^r_{A}, \mathcal{O}(-r-1))$$ is a perfect pairing of finitely generated free…
Sang Cheol Lee
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Pushforward of Pullback of Sheaves of Modules

If $f:X\rightarrow Y$ is a morphism of schemes and $M$ an $\mathcal{O}_Y$-module, then by definition $f^*M=f^{-1}M\otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$, where $f^{-1}$ denotes the inverse image of sheaves. Is it true that $$f_*f^*M\cong…
Topliner
  • 31
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Taylor series at singular points in algebraic geometry

There is a notion of a Taylor Series for regular functions on an affine variety X, given a set of generators $\{u_1,\cdots,u_n\} $of the maximal ideal $\mathfrak{m}$ at a point $P$ that form a basis for the cotangent space…
David Melo
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Cohomological implications of Twisting by Effective Nef Divisors

It is a standard result due to Serre that for a projective scheme $X$, $D$ an ample divisor on $X$, and $F$ a coherent sheaf, there exists an $N$ such that for all $n\geq N$, one has $H^0(X,F\otimes O_X(nD))\neq 0$ (or better yet, $F\otimes O_X(nD)$…
Shrugs
  • 1,458
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Base change of flat morphism preserve relative dimension

I have a flat morphism of relative dimension $d$. Is it true that a base change of this morphism is a flat morphism of the same relative dimension? We can assume that all schemes are of finite type over some field and equidimensional.
Galois group
  • 615
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Confusion between blow-up and relative projective space

Let $k$ be a field, $R = k[x,y]$ and $I = (x,y)$, so that $\operatorname{Spec}(R) = \mathbb{A}^2_k$ and $V(I) = \{0\} \subset \mathbb{A}^2_k$. By definition, the blow-up of the plane at the origin is the projective scheme associated to the Rees…
Eduardo de Lorenzo
  • 366
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How to show that $\operatorname{Hom}_X(f^{-1}\mathcal{G}, \mathcal{F}) = \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$?

Let $\mathcal{F}, \mathcal{G}$ be sheaves on topological spaces $X, Y$ respectively and $f: X \to Y$ a continuous map. How to show that $\operatorname{Hom}_X(f^{-1}\mathcal{G}, \mathcal{F}) = \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$? By…
LJR
  • 14,520
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Can automorphism groups grow upon shrinking a variety?

Let $U\subset X$ be a dense open of a smooth projective variety $X$ over the complex numbers. I am looking for examples of the following phenomena: Aut(X) is trivial, but Aut(U) is infinite. Or, slightly easier probably: Aut(X) is finite, but…
Harry
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