The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.
14
votes
8answers
3k views
(undergraduate) Algebraic Geometry Textbook Recomendations
What are the best algebraic geometry textbooks for undergraduate students?
6
votes
1answer
546 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 ...
8
votes
3answers
719 views
$\mathbb{A}^{2}$ 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?
20
votes
10answers
3k views
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, ...
5
votes
2answers
956 views
Local-Global Principle and the Cassels statement.
In a recent article I have read, i.e. " Lecture notes on elliptic curves ", Prof.Cassels remarks in page-110 that
There is not merely a local-global principle for curves of genus-$0$, but
...
4
votes
0answers
119 views
Relation between a generalization of Weil's abstract varieties and algebraic schemes
We would like to generalize this question when the base field $k$ is not necessarily algbraically closed.
We fix an algebraically closed field $\Omega$ which has infinite trancendence dimension over ...
9
votes
1answer
2k views
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 ...
9
votes
6answers
923 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 ...
19
votes
2answers
562 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 ...
13
votes
2answers
1k 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 ...
4
votes
1answer
163 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 ...
3
votes
1answer
241 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 ...
4
votes
3answers
153 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) ...
3
votes
1answer
490 views
Intuition and Stumbling blocks in proving the finiteness of WC group
After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle,
and coming to its ...
37
votes
2answers
3k 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 ...
