The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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26 views

Chow group of zero cycles of a product

I have been starting to learn about about chow groups. I don't know much yet, so hopefully the following is trivial: :-) For a smooth (projective, if you like) variety $X$ over a field $k$ I will ...
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1answer
50 views

Principal Bundle- Definition cum Exercise from “Geometry and Topology” by Bredon

The definition of fiber bundle can be found from here: Definition of Fiber Bundle Then Bredon defines Principal bundle in the exercise as follows: I am not able to show how K acts naturally on ...
3
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1answer
54 views

Complete intersections are connected

I have been stuck on this exercise in Vakil's notes (and moved on hoping it would come to me later), and it seems to be useful for other results (for example, when expressing curves as complete ...
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43 views

Fibers of toric morpisms

Let $f: X(\Delta_1) \to X(\Delta_2)$ be a toric morphism of toric varieties, and let $\sigma \subset \Delta_2$ be a cone, then for any point in the corresponding orbit $x \in O(\sigma)$ the fiber ...
2
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1answer
42 views

Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
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39 views

Chow Groups of $\mathbb{P}^n$

I'm trying to see that $A_k(\mathbb{P}^n)=\mathbb{Z}$ for all $k$. I am trying to do this with induction on $n$, by applying the excision sequence $$A_k(Y) \xrightarrow{i_*} A_k(X) \xrightarrow{j^*} ...
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74 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field (I would be happy working entirely over $\mathbb C$). Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices ...
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1answer
46 views

Unique line through two points in projective space

I'm trying to solve exercise I.3.15 in Hartshorne's Algebraic Geometry. The question starts as follows: Projection from a point: Let $ \mathbb{P}^{n } $ be a hyperplane in $ \mathbb{P}^{n+1 } $ ...
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1answer
70 views

Chern classes of tautological bundle over the Grassmannian G(2,4)

I've the following problem: I know how to calculate Chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Schubert calculus. If I am right, the Chern character should ...
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1answer
64 views

Learning Fibre Bundle from “Geometry and Topology” by Bredon

Bredon defines bundle projection in the following way: Then he defines Fibre Bundle The he Remarks about the condition 3. He says the map $\theta :U \rightarrow K $ exists. The only important ...
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34 views

Flex point on an elliptic curve

I have just started working through Pete Clark's elliptic curve notes, which are available here: http://math.uga.edu/~pete/EllipticCurves.pdf Early on, in section 2.1 on page 6, it is shown that the ...
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28 views

Exceptional locus of birational morphism is a divisor.

Let $f: V\to W$ be a proper birational morphism of smooth varieties, in a paper I'm reading the author claims that the exceptional locus of $f$ (i.e. the inverse image of the smallest closed set of ...
5
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1answer
53 views

Does pullback of schemes by monomorphism produce topological pullback?

Suppose I have scheme maps $X\to Z,Y\to Z$. It is not in general true that the fibre product $X\times_{Z}Y$ has the same underlying topological space (or even the same underlying set) as the fiber ...
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0answers
38 views

parametrization of a rational variety

Let $Y$ be a affine variety of dimension $d$ in affine space $\mathbb{A}^n$. Let the vanishing ideal of $Y$ be generated by $f_1,\dots,f_n \in A=k[y_1,\dots,y_n]$. Now suppose that $Y$ is rational. ...
5
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0answers
54 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
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0answers
38 views

Direct limit of $ \ \ \mathcal{D} = ((\mathcal{F} (U))_{U \in \mathcal{V}} \, \ (r \ : \ \mathcal{F} (U) \to \mathcal{F} (V))_{V \subset U}) $. [duplicate]

Let $ X $ be a topological space. Let $ \mathcal{F} $ be a sheaf on $ X $. Let $ U $ be an open subset of $ X $. Let $ \mathcal{V} $ the set of open neighborhoods of $ U $, which is the filter for ...
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18 views

What are the elements in $U/U(w)$?

Let $U$ be the maximal upper unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Then $$ U(w) = \{(a_{ij}) \in U: a_{ij} = 0, \text{ if } i<j, w^{-1}(i) < w^{-1}(j) \}. $$ My question ...
2
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1answer
61 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
4
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1answer
72 views

Algebraic geometry: difference between variety approach and scheme approach?

This would be an elementary question and sorry if this is duplicate one - but I could not find any satisfactory answer anywhere else. :-( I'm learning algebraic geometry not for its own but for the ...
2
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1answer
77 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
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1answer
50 views

Is the unique morphism from the empty scheme $\operatorname{Spec}((0))$ to some other scheme $X$ smooth?

This is a very pedantic question, but Is the unique morphism from the empty scheme $\emptyset = \operatorname{Spec}((0))$ to some other scheme $X$ smooth?
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1answer
135 views

Felix Klein's view on algebraic geometry

I think, as a first approach one would say that a geometry on a set $X$ is given by an inner product on $X$. Klein then links geometry to group theory by identifying a geometry on $X$ with a group of ...
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1answer
55 views

Relating ramification index of a map of curves to degree of vanishing

I am little confused about explicitly computing ramification index and relating it to degree of vanishing a polynomial. In particular I have the following example (when trying to prove the genus ...
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1answer
35 views

Associated points and reduced scheme

1) Let X is a locally Noetherian scheme without embedded point, show that X is reduced iff it is reduced at the generic points. 2) Let X is a locally Noetherian scheme (maybe has some embedded ...
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1answer
50 views

About Betti Numbers

I'm studying the book 'The Geometry of Syzygies' of David Eisenbud, but I'm having problem with the following step, in page 7 he says the we have a free resolution to the set of ten points in ...
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1answer
51 views

A question on smooth morphisms and 'pointwise' smooth morphisms

Let $X$ be a scheme, $x\in X$ a point and $f\colon \operatorname{Spec}(k(x))\to X$ the canonical morphism. Is $f$ always a smooth morphism? Now suppose $g\colon X\to Y$ is a scheme over some ...
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3answers
221 views

Example I.4.9.1 in Hartshorne (blowing-up)

Let $Y$ be the irreducible curve of $\mathbb{A}^2$ given by $y^2 = x^2(x+1)$. Let $t,u$ be homogeneous coordinates of $\mathbb{P}^1$. Then the total inverse image of $Y$ under the blowing-up $\phi: X ...
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0answers
41 views

Orientability in algebraic setting

I have the following (it can be very silly) question. Suppose I have a commutative algebra $A$ over a field $k$ of $char(k)=0$ which defines a $n$-dimensional smooth variety $X=Spec(A)$. What ...
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0answers
28 views

Sheafification part 2: Uniqueness of $\tilde{\varphi}$ and a formal consequence

I'd like to go on discussing the proof which I started to discuss here. The book says sending $(s_x)_x\in\tilde{\mathscr{F}}(U)$ to $(\varphi_x(s_x))_x\in\tilde{\mathscr{G}}(U)$ defines a morphism ...
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1answer
66 views

Sheafification: Show that $\tilde{\mathscr{F}_x}=\mathscr{F}_x$.

My today's question is about a proof of this book. More precisely we are talking about the proof of Prop. 2.24 on page 52. The book says that we have $\tilde{\mathscr{F}_x}=\mathscr{F}_x$ for all ...
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0answers
31 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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35 views

How to describe a homomorphism from a fundamental group to a finite group?

Let $S$ be a connected locally noetherian scheme with $s$ a geometric point of $S$. I read something like this: to give a surjective continuous homomorphism from $\pi_1(S, \bar{s})$ to a finite group ...
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1answer
31 views

Lines in CP^n and in CP^2

The question is In how many points a line in CP^n intersects CP^2?. By a line in CP, I mean a copy from CP^1. I have tried with a sytem of equations, (Because a line in CP^n is the zero locus of a ...
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0answers
93 views

Abstract algebraic geometry vs complex algebraic geometry

Sorry in advance if my question is not precise enough. I'm currently trying to study algebraic geometry on my own. I've started by trying to read Harsthorne and Liu's book. And i found it very ...
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1answer
53 views

Star in Serre duality

Why is there a dual bundle in Serre duality? Let $\mathcal E$ be a vector bundle over complex manifold $X$, without any metric anywhere, then one has a pairing $$(\Omega^{0,q} \otimes \mathcal E) ...
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1answer
20 views

Find a projectivity to create a graph.

I have the tetrahedron {xyzt=0} in projective space with homogeneous coordinate (x,y,z,t). I need to create a graph but the tetrahedron in affine coordinate is {xyz=0} and I can't visualize the ...
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1answer
25 views

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
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0answers
31 views

what does projective line of degree one mean?

I know what is projective line, but I'm confused about degree one. Can someone tell me what is "projective line with degree one"?
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27 views

Motive of Pfisterforms spectral sequence

In this famous paper http://www.math.uni-bielefeld.de/~rost/data/motive.pdf Rost constructs the motive of a Pfister-Form/Pfister-Quadric. In the last proof on page 13 he writes: "By a spectral ...
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1answer
56 views

Blowup of a (very) simple singularity

Take the action of $\mathbb{Z}_2$ on $\mathbb{C}^2$ given by $(-1) \cdot (z,w) = (-z,-w)$ and of course, $(1)\cdot (z,w) = (z,w)$. If you look at the resulting quotient space $\mathbb{C}^2 / ...
2
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1answer
97 views

The Zariski topology on $\operatorname{Spec} A$ as an intial topology

Given any commutative ring $A$ let $\operatorname{Spec} A$ be the space of prime ideals of $A$. Can we interpret the Zariski topology as an initial (or final) topology with respect to some canonical ...
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1answer
22 views

How to compute $U \cap wUw^{-1}$?

Let $U$ be the upper unipotent subgroup of of $GL_n$. It is said that $$ U \cap wUw^{-1} = \{ (a_{ij}) \in U \mid a_{ij}=0, i<j, w^{-1}(i) > w^{-1}(j) \}. $$ How to prove this? I try to compute ...
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1answer
44 views

Why notion of fundamental group is defined only over a connected scheme?

I went to different references on fundamental group on schemes. It is quite strange for me that the notion of fundamental group is only defined on connected scheme. Does anybody know why?
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1answer
48 views

Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces

Let $X$ be a smooth projective surface and $D_1, D_2$ be two divisors on $X$. Is it true that $D_1$ is linearly equivalent to $D_2$ if and only if $D_1$ is algebraically equivalent to $D_2$?
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25 views

Algebraic set in affine space

How to show that unitary group $U(n)\subset M_n(\mathbb C)=A_{\mathbb C}^{n^2}\cong A_{\mathbb R}^{2 n^2}$ is not complex algebraic set in complex affine space $A_{\mathbb C}^{n^2}$, but it is a real ...
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67 views

Atiyah & Macdonald's Introduction to Commutative Algebra, Exercise 8.5

The exercise asks the reader to prove that $X$ is a finite covering (i.e., the number of points of $X$ lying over a given point of $L$ is finite and bounded) of $L$, where the affine varieties $X$ and ...
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3answers
31 views

Empty set in projective space has negative dimension?

I've seen some notes online that show that a closed set in the projective space is nonempty by showing that it has dimension $ \geq 0 $. This seems to rely on the convention that the empty set has ...
2
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1answer
28 views

Number of Orbits of symmetric group acting on $(\mathbb{Z}/n)^{l}$

I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related ...
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0answers
25 views

ample anticanonical system implies regularity

I was having a look at Exercise V.21(1) of Beauville's "Complex Algebraic Surfaces", where it is asked to classify surfaces with ample anticanonical system. These are the surface $\mathbb{P}^1 \times ...
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1answer
68 views

Holomorphic line bundle with degree zero

I'm studying algebraic geometry and I need some help to understand the Riemann-Roch theorem. Let us consider a holomorphic line bundle $\xi$ over a Riemann surface $X$. The unique invariant of a ...