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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Dimension of subsets of nonsigular variety

Could you help me to prove this question? Do you have any idea? Let $X$ be a nonsingular variety and $Y \subseteq X$ is close and nonsingular . Then for any $x \in Y$ which $\dim Y_x=\dim X_x ...
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1answer
72 views

Finding the area enclosed by $px^4+qxy+ry^2+sy+t=0$

When $px^4+qxy+ry^2+sy+t=0\ (p,q,r,s,t\in\mathbb R)$ represents a simple closed curve on the $xy$ plane, can we represent the area enclosed by this curve by $p,q,r,s,t$? If yes, then how? Example 1 ...
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30 views

Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...
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2answers
90 views

geometric motivation for spaces with functions

Let $k$ be a field. A space with functions over $k$ is topological space X together with a family $O_X$ of k-subalgebras $O_X(U)\subseteq Map(U,k)$ for every open set $U$ that satisfy a) If ...
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59 views

Kähler differentials of the cuspidal cubic

I want to compute $\Omega^1_{A,\mathbb{C}}$ for $A = \mathbb{C}[X,Y]/(Y^2 - X^3)$, or more precisely, I want to show that the module of Kähler differentials is free of rank 2 at the origin, and free ...
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27 views

Well-definedness of the action of the structure group of a principal bundle on the total space.

Find the definition of a fiber bundle here- Definition of Fiber Bundle I am having difficulty in proving that the natural action of $K$ on $X$ is well-defined: Let us recall how does K acts on X ...
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73 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
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71 views

Invertibility of a polynomial map equivalent to a condition on the ideal generated by the coordinates?

Let $k$ be an algebraically closed field. Let $$F=(F_1,\ldots,F_n) \colon k^n \to k^n$$ be a polynomial map. I'm trying to understand the relation between the conditions: $F$ is invertible. ...
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552 views

Kähler differentials of affine varieties

I would like to gain some intuition regarding the modules of Kähler differentials $\Omega^j_{A/k}$ of an affine algebra $A$ over a (say - algebraically closed) field $k$. Let us recall the ...
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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 ...
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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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15 views

canonical divisor and self-intersection number

Let $X$ be a Chatelet surface over $\mathbb{Q}$ whose affine model is given by $y^2 + z^2 = P(x)$, where $\deg(P) = 4$, and let $K_X$ be the canonical divisor. How can I compute $K_X$ and $K_X^2$ ...
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23 views

Is $U/U(w) = U \cap w U^- w^{-1}$? [on hold]

Let $U$ be the maximal upper unipotent subgroup of $GL_n$ and $U^{-}$ maximal lower unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Is $U/U(w) = U \cap w U^- w^{-1}$? Thank you very much.
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56 views

Cardinality of variety

I'm trying to show that the cardinality of any variety of positive dimension is $ |k |$ where $k $ is the field being considered. This is part of exercise I.4.8 in Hartshorne's Algebraic Geometry: ...
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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 ...
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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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53 views

Regular elements of a module is open and dense

Let $\mathfrak{g}$ be an algebraic Lie algebra and $V$ be a $\mathfrak{g}$-module, then for each $v\in V$, define $\mathfrak{g}^v = \{x\in\mathfrak{g}:xv = 0\}$. Let $V_{reg}$ be the set of all $v$ ...
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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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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 ...
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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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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
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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136 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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212 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
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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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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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0answers
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 ...
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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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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. ...
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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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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 ...
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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 ...
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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
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
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 ...
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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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64 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
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1answer
36 views

Product of varieties in is a variety?

I know that the question may look similar to this: Is fibre product of varieties irreducible (integral)?, but I am forced by the context to use a different definition for variety. Definition. Let $K$ ...
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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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234 views

Associated primes and integral closure

Let $A$ be an integral domain which is finitely generated as a $k$-algebra and let $I\subset A$ be an ideal. Let $B$ be its integral closure (in the fraction field $\mathrm{Frac}\ A$) - in this case ...
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1answer
231 views

Working out the normalization of $\mathbb C[X,Y]/(X^2-Y^3)$.

I'm trying to identify the normalization of the ring $A := \mathbb C[X,Y]/\langle X^2-Y^3 \rangle$ with something more concrete. First, $X^2-Y^3$ is irreducible in $\mathbb C[X,Y]$, making $\langle ...
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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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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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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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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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1answer
88 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
3
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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 ...