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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Cohomology groups of a non-degenerate algebraic variety.

Let $X\subset\mathbb{P}^{n}$ be an algebraic variety. Let us suppose that $X$ is non-degenerate (it is not contained in any hyperplane of $\mathbb{P}^{n}$). I have read that (at least for curves) the ...
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A question about Weil restriction

Let $l/k$ be a finite Galois extension of fields of characteristic zero. Let $X$ be an affine scheme of finite type over $l$ and denote the Weil restriction by $\prod_{l/k} X$ (it exists in this ...
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Changes of Cohomologies in small resolution

Let $X$ be a singular complex variety of dimension $3$, whose singular locus is only a node! Suppose there exists a small resolution \begin{equation} \pi~:~\hat{X} \rightarrow X \end{equation} which ...
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29 views

Relative Frobenius Morphism of Finite Fields

Let $K$ be a finite field of characteristic $p$ and let $L$ be a finite extension of $K$. Then $L$ has an absolute Frobenius morphism which is given by the $p$th power map. Moreover, we have a map of ...
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whats wrong with this counterexample to closed subgroups of a Torus are a torus

In Cox Little and Schenck, one result that is cited in chapter two is that if $D_n$ is the $n-dimensional$ torus, and $H < D_n$ is a closed subgroup then $H$ is itself a torus. Let the underlying ...
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Arithmetically Cohen-Macaulay curve on a quadric

If $Y$ is a curve of bidegree $(a,b)$ on a smooth quadric surface $Q\subset \mathbb{P}^3$, how do we see that it is arithmetically Cohen-Macaulay (ACM, for short) iff $|a-b|\leq 1$? If (like me) ...
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Finite type assumption necessary for this property of very ample sheaves?

The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement: ...
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Four polytopes and their relation

Suppose we have four similar $A_{1},A_{2},A_{3},A_{4}$ polytopes in Euclidean Space. They are different and we know that $$ A_{1}\cap A_{2}=B_{1},~A_{2}\cap A_{3}=B_{2},~A_{3}\cap ...
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Weirestrass points in Principles of Algebraic Geometry.

On page 274 we have the gap values of $p\in S$ where $S$ is a Riemann surface, these are listed as follows: $$a_1 = 1 , a_2 = 2+\alpha_1 , \ldots , a_g = g+ \alpha_1 + \ldots \alpha_{g-1}$$ Now the ...
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35 views

If a linear transformation is defined over $F$, so is the kernel

Let $V$ be a vector space over a field $k$, and $F$ a subfield of $k$. An $F$-submodule $V_0$ of $V$ is called an $F$-structure if the natural $k$-linear map $V_0 \otimes_F k \rightarrow V$ is an ...
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Finding Singular points

Let $f = 3x^3 + 3x^2 − y^2 + z^2$, $g = 3x^2 + 4^x + 3y^2 + z^2$ be polynomials in $\mathbb{C}[x, y, z]$ and let $W = V(\langle f, g\rangle) ⊂ \mathbb{A}^3(C)$. By using the Jacobian matrix, find the ...
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62 views

My strange proof of the fact that $k[\mathbb P^n]=k$

Let $f\in k[\mathbb P^n]$, i.e. $f: \mathbb P^n \to \mathbb A^1$ be a regular function. My purpose is to show $f$ is constant. First $\mathbb A^1$is considered the subset $\{x=[x_0:x_1]\in \mathbb ...
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The sheaf (of stalks) of meromorphic functions, why don't we use a more natural definition?

If $A$ is a commutative ring with $1$, let's denote with $R(A)$ the set of regular elements of $A$. Let $(X,\mathcal O_X)$ be a locally Noetherian scheme, then the sheaf (of stalks) of meromorphic ...
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When a prime ideal is maximal differential ideal in a UFD?

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$? I know there are maximal ideals like ...
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45 views

Tensor product of coordinate rings corresponds to pullback

Here in Milne's notes on algebraic geometry, he proves that if $k$ is an algebraically closed field, and $A$ and $B$ are reduced finitely generated $k$ algebras, then $A \otimes_k B$ is reduced. (This ...
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Mirror Symmetry of Elliptic Curve

I'm a little bit unsure about the mirror symmetry statement for elliptic curves; specifically, how the flipping of the Kähler and complex moduli works. Perhaps I should say at the outset, the reason ...
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37 views

On the proof that one dimensional linear algebraic groups are either isomorphic to $\mathbb{G}_m$ or $\mathbb{G}_a$.

Let $G$ be a linear algebraic group of dimension one. The proof that I am looking at, in t.a springer's book (thm 3.4.9) proceeds by showing that $G$ must be either equal to its semisimple part ...
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Why are there no $\mathbb{R}$-valued points on a complex curve?

We say a $K$-valued point on a scheme is a map Spec$(K) \to S$, so in particular, a real valued point on the parabola $y = x^2$ should be a map Spec$(\mathbb{R}) \to $Spec$(\mathbb{C}[x,y]/(y-x^2))$. ...
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Embedding Complex Tori in Projective Space

When we talk about projectively embedding complex tori $\mathbb{C}^{g}/\Lambda$ (i.e in Lefshetz Embedding Theorem), what exactly do we mean by an embedding. Is it in the differential geometry sense ...
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A characterization of irreducible polynomials linear in one variable?

Let $p\in \mathbb{C}[x,y,z]$ be an irreducible polynomial such that for each $x,y\in (−1,1)$, there exists a unique $z_{x,y}\in \mathbb{C}$ suchthat $p(x,y,z_{x,y})=0$. Conjecture: There exist ...
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Differential of the Gauss map of an algebraic variety.

Let $X=V(F)\subset\mathbb{P}^{n}$ be a smooth irreducible hypersurface. Let us consider the morphism $$ \mathcal{G}:X\rightarrow \mathbb{P}^{N}, p\mapsto \left( \frac{\partial F}{\partial ...
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57 views

Chern class of ideal sheaf

Let $X$ be a smooth projective surface. Let $Z$ be a dimensional $0$ subscheme of length $l$. Suppose $I_Z$ is the ideal sheaf of $Z$. Then it claimed that $c_1(I_Z) = 0$ and $c_2(I_Z) = l$. (1)Why ...
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The same algebraic variety defined by different sets of polynomials

Let $\emptyset\neq X\subset\mathbb{P}^{n}$ be an algebraic variety such that $$ X=V(F_{1},\ldots,F_{m}) $$ for certain linearly independent homogeneous polynomials $F_{1},\ldots,F_{m}\in ...
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34 views

Show that $D$ is not linearly equivalent to any other effective divisor

Let $C$ be a nonsingular quartic, $P_1,P_2,P_3 \in C$. Let $D=P_1+P_2P_3.$ Let $L$ and $L'$ be lines such that $L \bullet C= P_1+P_2+P_4+P_5$ and $L'\bullet C= P_1+P_3+P_6+P_7.$ Suppose these seven ...
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51 views

Ellipse's farthest point to another point

I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ...
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30 views

Moving Line Segment Problem part 2

This question is related to a question I asked a while ago here on math.stackexchange: Moving Line Segment Problem The rules for how the line segment can be moved are the same: The endpoints must ...
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1answer
82 views

Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
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Cartier divisor and its support.

(1) Let $X$ be a Noetherian scheme of dimension 1 over a field $k$ and $supp(D)$ denote the support of a $Cartier$ divisor $D$ on $X$. Let $S\subseteq supp(D)$ be consists of closed points of ...
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Quotient Sheaves

Let $X$ be a ringed space, and $J$ be a sheaf of ideals of the structure sheaf. Define, $Y = \{x\in X ~ | ~ J_x \not = \mathcal{O}_x\}$, this is a closed set. We have an inclusion $i:Y\to X$. Is there ...
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Schemes not determined by morphisms from one object

I was reading a bit about the functor of points of a scheme, and it was mentioned that there does not exist a scheme $Y$ so that Hom$(Y,X)$ determines the points $X$ for all schemes $X$. This is in ...
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Universal property of relative proj in simple situations?

On the stacks project they have a complicated universal property for Proj: http://stacks.math.columbia.edu/tag/01NS Presumably, at least some of the complexity there is due to the insistence on ...
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When exactly is a compact complex manifold algebraic?

It is well known that a necessary and sufficient condition for a compact Kähler manifold $\mathcal{X}$ to be a projective algebraic variety is that it admit a positive holomorphic line bundle $L ...
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Pullback of correspondence on Chow-Rings

For a smooth, projective variety $X$, one gets a pull back $f^*$ $f^*:$ CH$^*(X\times X) \rightarrow $ CH$^*(X)$, of the map $f: X \rightarrow X \times X$. Let $X_3$ be a quadric form, for ...
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Representation of an effective Cartier divisor: is there an imprecision in Liu's book?

I will use the notation of Liu's book which is also the standard one. If you don't feel comfortable with it please ask more information in the comments Let $X$ be a scheme, and consider a Cartier ...
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Complex manifold with subvarieties but no submanifolds

Note, I have now asked this question on MathOverflow. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. For example, generic tori of ...
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76 views

Section of a coherent sheaf vanishing outside a point

I need some help in understanding an argument, probably basic, about coherent sheaves, which I've read in a paper, and as far as I understand can be described as follows: Let $\mathcal F$ be a ...
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Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in ...
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Question regarding projective coordinate transformation

While reading Kunz's commutative algebra book, I came across a statement I can't understand. First, let me define the notations. Let $L/K$ be extension of fields, and let $\mathbb{P}^n (L)$ denote ...
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Is this result about the defining ideal true?

I am trying to generalize a result whose precise statement is the following: Let $X$ denote a set of $d+1$ points of $\mathbb P^{d}$ and the points are in linearly general position. Then $I_X$, ...
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28 views

Using Riemann-Roch

I have canonical divisor $K$ of curve $\mathbb{P}^1$, and I would like to find $l(2K)$, $l(3K)$ and $l(-K)$ using Riemann-Roch theorem. I know that $g=0$ in this case, so $deg(K)=2\cdot 0-2=-2$. ...
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How to determine the ideal defining $\overline{f(Z)}$ where $f$ is a regular map of affine variety.

Let $f:X\to Y$ be a regular map of affine varieties. Let $Z$ be a closed set of $X$ and $I$ be the radical ideal defining $Z$. My question is how to determine the ideal defining $\overline{f(Z)}$? Can ...
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Exercise I-5 of Eisenbud-Harris

I have just started learning about schemes, so please do not be too harsh with me. I am trying to do Exercise I-5 of Eisenbud-Harris, "The geometry of schemes". Suppose $X$ is the topological space ...
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What is the correspondence between primary decomposition and algebraic geometry? [duplicate]

What is the correspondence between primary decomposition and algebraic geometry?
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A morphism from a projective curve $X$ to a curve $Y$ is either constant or surjective

How to prove with riemann roch "A morphism from a projective curve $X$ to a curve $Y$ is either constant or surjective, if it is surjective then $Y$ must be projective"?.
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Morphism smooth over the function field. What does it mean?

Look at the following lines I found in the book "Moriwaki - Arakelov geometry" (beginning of chap. 4): Let $S$ be a connected Dedekind scheme with function field $K$. Let $\pi:X\to S$ be a ...
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Thus the decomposition of algebraic sets into irreducible ones corresponds to writing a radical ideal as an intersection of prime ideals

I read this assertion in Irena Swanson's Thesis:$\qquad$ $\qquad$ $\qquad$ $\qquad$ $\qquad$ $\qquad$$\qquad$$\qquad$$\qquad$$\qquad$"In the standard correspondence between algebraic sets and ideals, ...
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How are non-homogenous elliptic curves projective varieties?

So if I am given an elliptic curve such as $Y^2Z=X^3$ then I see how it can be realized as the projective variety $Proj(k[X,Y,Z]/(Y^2Z-X^3))$. But, given an elliptic curve like $Y^2 = X^3 + X$, then ...
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Sheaf associated to a Cartier divisor

This question is motivated by a construction, unclear for me, related to Cartier divisors. But, in the end it can be reduced to a question involving only sheaves on topological spaces. Let $X$ be a ...
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42 views

Is the restriction of a finite map of affine varieties also finite?

If $f:X\to Y$ is a dominant(i.e.$f(X)$ is dense in $Y$) regular map of affine varieties, then $f$ is called a finite map if $k[X]$ is integral over $f^*k[Y]$. My question is: if $Z\subset X$ is a ...
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41 views

On the anti-equivalence of affine schemes with commutative rings

There is an equivalence $\mathbf{Aff}\simeq \mathbf{CRing}^{\text{op}}$ between the category of affine schemes and the category opposite to the category of all commutative rings. If we instead ...