# Tagged Questions

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. ...

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### (undergraduate) Algebraic Geometry Textbook Recommendations

What are the best algebraic geometry textbooks for undergraduate students?
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### $\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
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### Divisor — line bundle correspondence in algebraic geometry

I know a little bit of the theory of compact Riemann surfaces, wherein there is a very nice divisor -- line bundle correspondence. But when I take up the book of Hartshorne, the notion of Cartier ...
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### Best Algebraic Geometry text book? (other than Hartshorne)

Lifted from Mathoverflow: I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best. Then what might be the 2nd best? It can be a book, preprint, online lecture note, ...
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Krull dimension of a ring $R$ is the supremum of the number of strict inclusions in a chain of prime ideals. Question 1. Considering $R = \mathbb{C}[x_1, x_2, x_3, x_4]/\left< x_1x_3-x_2^2,x_2 ... 3answers 24k views ### Why study Algebraic Geometry? I'm going to start self-stydying algebraic geometry very soon. So, my question is why do mathematicians study algebraic geometry? What are the types of problems in which algebraic geometers are ... 1answer 839 views ### Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald's Introduction to Commutative Algebra) stating that if$\mathfrak{a}$is a decomposable ideal of$A$(a ... 3answers 636 views ### The vanishing ideal$I_{K[x,y]}(A\!\times\!B)$is generated by$I_{K[x]}(A) \cup I_{K[y]}(B)$? Let$K$be a field,$x=(x_1,\ldots,x_m)$,$y=(y_1,\ldots,y_n)$,$A\!\subseteq\!\mathbb{A}^m_K$,$B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) \... 1answer 915 views ### Existence of valuation rings in an algebraic function field of one variable The following theorem is a slightly modified version of Theorem 1, p.6 of Chevalley's Introduction to the theory of algebraic functions of one variable. He proved it using Zorn's lemma. However, Weil ... 2answers 1k views ### How to compute the topological space of fibered product of schemes? I know that the topological space of fibered product of schemes is generally distinct to the usual Cartesian product of toplogical spaces of schemes. Then how can we compute the top. sp. of fibered ... 3answers 3k views ### Dominant morphism between affine varieties induces injection on coordinate rings? Here are the definitions that we use for this problem: A morphism \varphi : X \to Y between two varieties is said to be dominant if the image of \varphi is dense in Y (c.f. Hartshorne exercise ... 1answer 827 views ### Tensor product of reduced k-algebras must be reduced? Let A, B be two reduced k-algebras. Then if an element of the form$$\sum a_{i}\otimes b_{j}$$is nilpotent, we can compose it with any k-homomorphism f from A to k to get a homomorphism ... 6answers 5k views ### Why Zariski topology? Why in algebraic geometry we usually consider the Zariski topology on \mathbb A^n_k? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ... 2answers 6k views ### Path to Basics in Algebraic Geometry from HS Algebra and Calculus? In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ... 6answers 4k views ### Reference for Algebraic Geometry I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is ... 2answers 2k views ### Tensor product of domains is a domain I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let A and B be k-algebras, which are ... 1answer 479 views ### Is the number of prime ideals of a zero-dimensional ring stable under base change? Let A be a zero-dimensional ring of finite type over a field k and let X= \textrm{Spec} \ A be its spectrum. Note that X is a finite set. Suppose that k\subset K is a finite field extension ... 1answer 280 views ### Modules over a functor of points I have a question on the ''functor of points''-approach to schemes and \mathcal{O}_X-modules. Please let me first write up a defintion. Let Psh denote the category of presheaves on the opposite ... 4answers 1k views ### Picard group of product of spaces Suppose X,Y are varieties over an algebraically closed field k. Can we compute \operatorname{Pic}(X \times_k Y) in terms of \operatorname{Pic}(X),\operatorname{Pic}(Y)? It seems that \... 2answers 534 views ### The bijection between homogeneous prime ideals of S_f and prime ideals of (S_f)_0 It is well-known that if S is a graded ring, and f is a homogeneous element of positive degree, then there is a bijection between the homogeneous prime ideals of the localization S_f and the ... 3answers 2k views ### Are “n by n matrices with rank k” an affine algebraic variety? Identify the set of all complex n by n matrices with \mathbb{C}^{n^2}. We say a subset S \subset \mathbb{C}^{n^2} is an affine algebraic variety if S is the common zero set of a collection (... 1answer 887 views ### Number of points in the fibre and the degree of field extension Let X,Y be varieties over \mathbb{C}, k(X), K(Y) be function fields of X, Y. Suppose \pi: X \to Y is a dominant, \textit{injective}\ morphism, why the degree of the function field ... 1answer 694 views ### Prove that k[x,y,z,w]/(xy-zw), the coordinate ring of V(xy-zw) \subset \mathbb{A}^4, is not a unique factorization domain I want to show that k[x,y,z,w]/(xy-zw), the coordinate ring of V(xy-zw)\subset\mathbb{A}^4, is not a unique factorization domain. Morally, all we need to do is find some nonzero element that ... 2answers 609 views ### Generating Pythagorean triples for a^2+b^2=5c^2? Just trying to figure out a way to generate triples for a^2+b^2=5c^2. The wiki article shows how it is done for a^2+b^2=c^2 but I am not sure how to extrapolate. 3answers 681 views ### A complex algebraic variety which is connected in the usual topology Hartshorne wrote in his book's Appendix B that it can be easily proved that a complex algebraic variety is connected in the usual topology if and only if it is connected in Zariski topology. How can ... 5answers 3k views ### Why should I care about adjoint functors I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ... 2answers 2k views ### How do different definitions of “degree” coincide? I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do ... 2answers 4k views ### Elliptic Curves and Points at Infinity My undergraduate number theory class decided to dip into a bit of algebraic geometry to finish up the semester. I'm having trouble understanding this bit of information that the instructor presented ... 5answers 2k views ### geometric motivation for negative self-intersection consider the blow-up of the plane in one point. Let E the exceptional divisor. We know that (E,E)=-1. Which is the geometrical reason for which the auto-intersection of E is -1? In general ... 5answers 2k views ### Is there a way of working with the Zariski topology in terms of convergence/limits? As someone who is very fond of analysis, I feel most comfortable working in topological spaces via the notion of convergence of sequences (or nets, in infinite-dimensional Banach spaces, etc.). In ... 2answers 2k views ### Slick proof the determinant is an irreducible polynomial A polynomial p over a field k is called irreducible if p=fg for polynomials f,g implies f or g are constant. One can consider the determinant of an n\times n matrix to be a polynomial in ... 6answers 2k views ### Algebraic Geometry Text Recommendation I need to learn about Algebraic Geometry (perhaps from in the context of finite fields) and am looking for a recommendation for a text. Now, I've already done a search and checked out what was ... 2answers 874 views ### on the adjointness of the global section functor and the Spec functor In fact, it is an exercise on Hartshorne, Ex 2.4 to the second chapter of it (P79): let A be a ring and (X,\mathcal{O}_X) be a scheme. Given a morphism f:X\longrightarrow \operatorname{Spec} A,... 1answer 388 views ### Geometrical interpretation of I(X_1\cap X_2)\neq I(X_1)+I(X_2), X_i algebraic sets in \mathbb{A}^n Edit: I should point out that I'm working over an algebraically closed field k. Let X_1,X_2\subset\mathbb{A}^n be affine algebraic sets. Show that I(X_1\cap X_2)=\sqrt{I(X_1)+I(X_2)}. Show ... 2answers 6k views ### Decomposing an Affine transformation An affine transformation is composed of rotations, translations, scaling and shearing. In 2D, such a transformation can be represented using an augmented matrix by$$ \begin{bmatrix} \vec{y} \\ 1 \... 3answers 1k views ### When do equations represent the same curve? Suppose we have two sets of parametric equations$\mathbf c_1(u) = (x_1(u), y_1(u))$and$\mathbf c_2(v) = (x_2(v), y_2(v))$representing two 2D planar curves. When I say "2D planar curves" I mean ... 3answers 686 views ### Krull Dimension of a scheme Can someone give a hint or a solution for showing that a scheme has Krull dimension$d$if and only if there is an affine open cover of the scheme such that the Krull dimension of each affine scheme ... 3answers 914 views ### Is the set of closed points of a$k$-scheme of finite type dense? Let$k$be a field. Let$X$be a scheme of finite type over$k$. We denote by$X_0$the set of closed points of$X$. Is$X_0$dense in$X$? Motivation See my comment to Martin Brandenburg's answer to ... 1answer 244 views ### Is there a “geometric” interpretation of inert primes? I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ... 2answers 1k views ### Conditions such that taking global sections of line bundles commutes with tensor product? Let us work with projective algebraic varieties over$k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties. Of course it is not in general true that given two line bundles$L, ...
Let $k$ be an algebraically closed field and $X,Y$ varieties (i.e. integral, separated schemes of finite type over $k$). Is the fibre product $X \times_k Y$ necessary irreducible or integral? I ...