commutative algebra - A guide to Algebraic Geometry

I have completed one semester course on Commutative Algebra and Riemann Surfaces, and currently I am trying to read Algebraic Geometry. While reading from different books I feel that I must need a direction such that I can cover much topics in a semester. That is to say, what are the fundamental results or topics in Algebraic Geometry that I should read in a one semester course. Besides this I feel that I should follow some text books or some well written lecture notes as a prepratory metarial. It will be helpful to me if someone guide me by sa...Read more

algebraic geometry - Vertex and axis of $n$-dimensional paraboloid

Consider a surface defined by the form$$ x^\text{T}Ax+b^\text{T}x+c=0, $$where $A\in\mathbb{R}^{n\times{n}}$ is non-zero symmetric positive semi-definite, $b\in\mathbb{R}^n$ and $c\in\mathbb{R}$. Suppose that $\det(A)=0$, and that$$ \det\begin{pmatrix}A&b\\b^\text{T}&c\end{pmatrix}\neq0. $$Question 1: Does the set $S=\{x:x^\text{T}Ax+b^\text{T}x+c=0\}$ have a well-defined notion of vertex and axis of symmetry (analogous to a parabola in 2D)?Question 2: If so, is there a way to express the vertex and axis of symmetry in terms of the data...Read more

algebraic geometry - Single point perspective tunnel movement function

Imagine you are in a tunnel which is a perfect cylinder placed horizontally. The tunnel is formed by an infinite number of segments with same girth (4 meters diameter) and a fixed length of 20 meters. Your point of view is in the center of the cylinder section. You are travelling through the tunnel with a constant speed, 5 m/s. In between the segments there's a distinctive groove that is visible to you, lying in a plane that is perpendicular to your viewing direction vector, your travelling direction vector and the tunnel wall. The only way th...Read more

algebraic geometry - What are the differences among an affine variety, a vector space, and a projective variety?

What are the differences among an affine variety, a vector space, and a projective variety?Are there some nice examples to explain this? Edit: For example, what is the difference between the following: $k^n$ ($k$ is the ground field) as a vector space, and $k^n$ as an affine variety (what are the equations satisfied by $k^n$)? Why is $k^n$ not a projective variety? Why do we need to introduce projective varieties (do they have some advantages over affine varieties)? Why are $GL_n$ and $SL_n$ algebraic varieties? (What equations are satisfied by...Read more

algebraic geometry - Why are projective spaces and varieties preferable?

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. In a more elementary stage of mathematics, when we try to find solutions to given equations, in $\mathbb R$ or $\mathbb C$, are we thinking in a more affine way? I really find it easier to understand things or to draw pictures in affine spaces, since it "looks" much more like the space we are living in. So why do people, for example Hartshorne, always transform problems in affine spaces to problems in projectiv...Read more

algebraic geometry - Extension by zero not Quasi-coherent.

Hartshorne's Example 5.2.3 in Chapter 2 states that if $X$ is an integral scheme, and $U$ is an open subscheme with $i:U \rightarrow X$ the inclusion, then if $V$ is any open affine not contained in $U$, $i_{!}(\mathcal{O}_U) \mid_{V}$ will have no sections over $V$. But it will have non-zero stalks, and so cannot come from a module on $V$ and so is not be quasi-coherent.I get everything but for the fact that the extension by zero will have no sections over $V$. My main problem is that Hartshorne states that $i_{!}(\mathcal{O}_U)$ is the sheaf ...Read more

algebraic geometry - Sufficient condition for a function to be a bijection

We want to prove two sets $A$, and $B$ have the same cardinality. Assume we have found a function $f:A\to B$, and a function $g:B\to A$, with $f\circ g=id$. Can we conclude that $f$ is bijective?Context: I am reading a proof that the morphisms $\phi:V\to W\subset k^m$ of algebraic varieties, over an algebraically closed field $k$, are in bijective correspondence with the homomorphisms of their respective coordinate rings as $k$-algebras. To each morphism, we assign its pullback. Conversely, to each homomorphism $a:k[W]\to k[V]$ we assign the mo...Read more

algebraic geometry - Points on the "conic" $\mathbb{R}[x, y] / (x^2 + y^2 + 1) $

On exercise 2.1.3 of Qing Liu's algebraic geometry book, which I'm self studying together together with Vakil's notes, he asks us to prove the variety defined by $x^2 + y^2 + 1$ over the real field looks like the projective plane minus a point.The later more abstract parts of the exercise I was able to do. However the first step, where he asks us to show that the maximal ideals in the quotient ring must contain single variable quadratic polynomials $x^2+ax+b$ and $x^2+cx+d$, I haven't been able to.I suspect we can use the fact that quotient by ...Read more

algebraic geometry - Closed points on varieties

I consider a variety over a field $k$, i.e. an integral separated scheme $X$ of finite type over $k$.One knows by the Nullstellensatz that any closed point on $X$ is a $\bar k-$ rational point (where $\bar k$ denotes the algebraic closure of $k$)as its residue field is finite over $k$.I know wonder what one can say about the relation between the closedness of a point and its residue field. E.g. it wont hold that any $\bar k-$rational point is closed, but can one say something similar? Or how can one characterize the closed points?And does the ...Read more

algebraic geometry - $F$-rational points of variety/alg. group in Springer

I am confused about something on page 6 in Springer's Linear Algebraic Groups.Setup:$k$ is an algebraically closed field$X$ is a closed set in $k^n$.$I(X) = \langle f\in k[T_1, \dots, T_n] : f(v) = 0\; \forall v\in X \rangle$$A = k[X] = k[T_1 ,\dots , T_n] / I(X)$$F$ is a subfield of $k$.I understand that an $F$-structure on $X$ is a $F$-subalgebra $A_0 = F[X]$ of $k[X]$ such that $A_0$ is of finite type and $k\otimes_F A_0 \simeq k[X]$.Question: What are the $F$-rational points of $X$? Springer says that it is all $F$-algebra homomorphisms fr...Read more