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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2
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1answer
36 views

Can the cohomology of the Grassmannian identified with the cohomology of a specific dense open subvariety?

Let $(\mathbb{C}^{2p},Q)$ be a $2p$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form $Q$ where $p\geq 3$. Let $l\leq p-2$. You may assume that $l$ is odd if this ...
2
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0answers
38 views

Can motivic cohomology say anything about real points?

Let $X$ and $Y$ be two smooth schemes defined over $\mathbf Z$. Suppose that we have a scheme morphism inducing isomorphisms on motivic cohomology groups $\mathrm H^p(Y,\mathbf Z(q)) \simeq \mathrm ...
0
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0answers
19 views

Sheafs of modules on Proj S

This is a question related to Proposition $5.12$, Part $(C)$ of Hartshorne (Page 117). Let $S,T$ be graded rings, and let $X = Proj S$, $Y = Proj T$. Assume that $S, T$ are generated respectively by ...
0
votes
0answers
6 views

Weighted and Generic vector

I am writing a project about tropicalization of toric varieties and in my work I come up with 2 expressions I do not know the meaning of. A weighted vector This is an example of where it is used. ...
0
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0answers
5 views

Relationship between the averages and derivatives of concave functions

I have two functions $S^m(K)$ and $S^f(K)$, such that $S^m(K)$ > $S^f(K)$ $\forall K>0$. Also, $S_K(K) >0$ and $S_{KK}(K) <0$, making the functions increasing and concave. Also, ...
0
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0answers
39 views

Comprehensive Mathematics References/Textbooks (Like Bourbaki's Elements, or the Stacks Project)

Are there any comprehensive mathematics reference/textbooks that could be considered somewhat like a modern version of Bourbaki's Elements? "Comprehensive" here could refer to a single area of ...
3
votes
1answer
82 views

Induced ring homomorphism by map on spectra

I know that when we have a ring homomorphism $$ \phi: A\rightarrow B$$ this induces continuous map between the spectra $$ \phi': \operatorname{Spec} B\rightarrow \operatorname{Spec}A$$ which maps $p$ ...
0
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0answers
32 views

Computing the restriction $(\Lambda^2T_{\mathbb{P}^n})_{|L}$ for line in $\mathbb{P}^n$

From the Euler sequence $$0\to\mathcal{O}_{\mathbb{P}^n}\to V\otimes\mathcal{O}_{\mathbb{P}^n}(1)\to T_{\mathbb{P}^n}\to0$$ it is easy to deduce that ...
3
votes
2answers
47 views

Why Brauer-Severi varieties over number fields satisfy the Hasse principle?

If I understand correctly, a Brauer-Severi variety is supposed to be a variety $X/k$ such that it becomes isomorphic over $\overline{k}$ to some projective space $\mathbb{P}_{k}^{n}$. When $k$ is a ...
1
vote
1answer
41 views

Lie Algebra of a connected simple linear algebraic group

Let $G$ be a linear algebraic group and $A=K[G]$ (K is a field of characterstic 0) be the coordinate ring of $G$. In Humphreys, the Lie algebra of $G$ is defined as the space of left invariant ...
2
votes
1answer
46 views

Linear Algebraic Group acting on the co-ordinate ring

Let $G$ be a linear algebraic group and let $C[G] = C[x_1,...,x_n] / I(G)$ denote the coordinate ring of $G$. (Note that $I(G)$ is the ideal containing all those polynomials in $C[x_1,...,x_n]$ of ...
0
votes
1answer
23 views

Automorphism of the coordinate ring of a linear algebraic group

Let me define a linear algebraic group first of all: A linear algebraic group is an affine variety (zeroes of a set of polynomials in $n$ variables with coefficients in a field $C$) such that the ...
0
votes
1answer
50 views

Possible subbundles on $\mathbb{CP}^1$

I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why? We know that the ...
0
votes
2answers
27 views

Coordinate ring of $GL_2$.

Let $GL_2$ be the group of all 2 by 2 invertible matrices over a field $K$. Let $x_{ij}$ be the function on $GL_2$ such that $x_{ij}(a) = a_{ij}$ for $a = (a_{ij}) \in GL_2$. Is the coordinate ring ...
3
votes
1answer
55 views

Smooth affine plane curve with non-trivial cotangent sheaf?

Question: Let $A = \mathbb C[x,y]/(f)$ be a non-singular plane curve. Under what conditions is the module of Kahler differentials $\Omega_A^1$ (over $\mathbb C$) a free module? I am not sure what ...
-1
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0answers
51 views

Zeros of homogeneous polynomials [closed]

When can we say that an affine variety $V(P1,...Pr)$ is non empty, where $P_i$ are homogeneous polynomials over $C[X1,...,Xn]$ ? I'm looking for the condition about $r$: when $r<n$ and especially ...
1
vote
1answer
28 views

Explain Multidegree of a polynomial

Definition (The book Ideals, Varieties and Algorithms, Cox et al, pages 59-60 on 2008 edition): Let $f=\sum_a a_a x^a$ be a nonzero polynomial in $k[x_1,\ldots, x_n]$ and let > be a monomial ...
3
votes
1answer
45 views

If the $m-1$ first derivatives of a rational function vanish at a point, does the function have a zero of order $m$ at that point?

Let $C\subseteq\mathbb{P}^{2}$ be a projective smooth algebraic curve, and let $$ \alpha:K(C)\rightarrow K(C) $$ be a derivation, i.e. $\alpha$ is a $K$-linear map such that $$ ...
2
votes
1answer
50 views

Complex Elliptic Surfaces without sections

Is there a description of smooth complex projective surfaces without sections? While working on a problem a surface $X$ showed up with the following property: it is a non-ruled surface that has an ...
2
votes
1answer
33 views

Computing the restriction $T_{\mathbb{P}^3|X}$ for twisted cubic in $\mathbb{P}^3$

Let $i:X=\mathbb{P}^1\to\mathbb{P}^3$ be a twisted cubic given by the embedding $(u:v)\mapsto(u^3: u^2v: uv^2: v^3)$. How to compute $T_{\mathbb{P}^3|X}$?
0
votes
1answer
40 views

Scheme morphism properties that aren't stable under taking triangles?

Let $\mathcal{P}$ be the collection of properties of morphisms of schemes that satisfy the following conditions: Stability under arbitrary pullbacks Stability under composition There's a nice list ...
1
vote
1answer
32 views

Ext sheaves of Ideal Sheaf

Let $X$ be a smooth variety (over $\Bbb{C}$) and $Y\subset X$ a smooth codimension one irreducible subvariety. How to compute the Ext sheaves $\mathrm{Ext}^i(I_Y,O_X)$ ? In particular, when $\dim ...
1
vote
0answers
30 views

Remove a non-base point from a linear system of a curve, How the dim changes?

Suppose X is a smooth projective curve. And D is an effective divisor. If P is a non-base point of the linear system |D|, what can we say about dim|D-P| ? I was reading Hartshorne Chapter IV, section ...
2
votes
1answer
23 views

In what ways can one check if the intersection of hypersurfaces in weighted projective space is smooth?

Let $\Bbb P^n_q=\Bbb P(q_0,\dotsc, q_n)$ be a weighted projective space and let $f_1,\dotsc, f_k$ be $q$-homogeneous polynomials of degrees $d_1,\dotsc,d_k$ respectively. Let $X$ be defined as the ...
1
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0answers
68 views

Research ideas in Homological algebra

I am planning to focus my research on Homological Algebra and related fields. I am on my first year, first semester and currently pursuing courses on Homological Algebra, Algebraic Geometry (first ...
2
votes
1answer
41 views

geometric meaning of blowing up the affine space at a line

The blow-up $\tilde{X}$ of $\mathbb{A}^3$ at the line $\ell: x_1=x_2=0$ is by definition the closure in $\mathbb{A}^3 \times \mathbb{P}^1$ of the graph of the function $f: \mathbb{A}^3 - \ell ...
3
votes
0answers
94 views

Deformation theory introduction without unnecessary machinery

I would like to find an introduction (book, article and/or lecture course) in deformation theory that does not use unnecessary machinery (for example, schemes instead of complex varietie, or ...
5
votes
0answers
31 views

Arithmetic genus of complete intersection going wrong

I want to compute the arithmetic genus of the complete intersection of two quadrics in $\mathbb{P}^3$. I know the adjunction formula is one way of doing this, but I am keen to understand where my ...
1
vote
1answer
27 views

Division with remainder in $K[x,y]$

Let $K$ be a field and $f\in K[x,y]$ such that $a:=\frac{\partial f}{\partial x}(0,0)\not=0$. Let $b:=\frac{\partial f}{\partial y}(0,0)$. In a lecture of mine, it was claimed now that we can divide ...
1
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0answers
24 views

If $\pi : Proj(T_*) \to Proj(S_*)$ is induced by some $S_* \to T_*$, does some tensor product describe $\pi^*(\widetilde{M_*})$

If $\pi : Proj(T_*) \to Proj(S_*)$ is induced by some $S_* \to T_*$, does some tensor product describe $\pi^*(\widetilde{M_*})$? $\pi^*(\widetilde{M_*}) =_? T_* [\otimes]_{S_*} M_*$?, where ...
0
votes
1answer
41 views

Spec $\mathbb{Z}[X]$ in Mumford's Red Book

In Chapter II of Mumford's Red Book he discusses Spec $\mathbb{Z}[X]$, and states that $V((p))$ is a copy of the affine line over $\mathbb{Z}/p\mathbb{Z}$. How is this so? Here is what I understand so ...
3
votes
1answer
35 views

Is Proj of the tensor product of two graded $A$-algebras isomorphic to the fiber product?

Some sanity checks: If $S_*$ and $T_*$ are two $A$-algebras, then is $Proj(S_* \otimes_A T_*) \cong Proj(S_*) \otimes_A Proj(T_*)$? Here the tensor product means $(S_* \otimes T_*) = \oplus_{n \geq ...
0
votes
0answers
29 views

How to find a vector in a hyperplane

I have a hyperplane (H) passes every corner points(one coordinate gets its maximum while others 0), such as ($u_1$,0,...,0), (0,$u_2$,0,...,0),....,(0,...,$u_n$). By linearity, points x in H satisfy ...
1
vote
2answers
73 views

Proving that $A^{T}A = M$ for all symmetric complex matrices $M$.

I am trying to prove that the map $M_{n}(\mathbb{C}) \rightarrow $ nxn Symmetric matrices $\cong \mathbb{C}^{n(n+1)/2}$, given by $A \rightarrow A^{T}A$ is surjective. I have shown that this map maps ...
0
votes
1answer
50 views

Question about geometric interpretation of modules

I would like to understand the accepted answer to this MO question about the geometric interpretation of modules. In particular, I would like clarification on the following excerpt. Let $R$ be the ...
1
vote
0answers
30 views

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?

If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$? This gives a graded ring, but it is not quite the tensor product since we ...
2
votes
1answer
53 views

Composition of morphisms of locally ringed spaces

I have a specific question about defining the composition in (locally) ringed spaces. The definition I had formulated myself while reading Hartshorne, since he conveniently neglected to suggest any ...
3
votes
2answers
42 views

If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} ...
2
votes
0answers
62 views

What is a morphism of Tannakian categories?

In this question, a Tannakian category over $k$ is a $k$-linear rigid symmetric monoidal tensor category, with the property that it has a fibre functor to $\mathbf{Vect}_\ell$ for some field extension ...
1
vote
2answers
44 views

Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes. My only geometric intuition for sheaf cohomology is via ...
1
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0answers
20 views

Extending a $\mathcal{B}$-sheaf via inverse limits vs stalks.

Let $X$ be topological space and $\mathcal{B}$ a basis. A $\mathcal{B}$-sheaf is a contravariant functor $\mathcal{O}_{\mathcal{B}}$ from $\mathcal{B}$ to the category of abelian groups that satisfies ...
2
votes
1answer
37 views

Calculating the ring of invariants for the action of $\mathbb C^*$ on $\mathbb C^2\setminus \{0\}$

Let $\mathbb{C}^*=\mathbb C\setminus\{0\}$ act on $\mathbb C^2\setminus \{0\}$ by scalar multiplication, where $\mathbb C^2=\operatorname{Spec}(\mathbb C[x_0,x_1])$. Then $\mathbb C^2\setminus ...
1
vote
1answer
27 views

Finding an irreducible polynomial

Suppose k is infinite. Then the irreducible algebraic subsets of $A^2(k)$ are: $A^2(k)$,$\emptyset$, points, and irreducible plane curves $V(F)$, where $F$ is an irreducible polynomial and $V(F)$ is ...
5
votes
1answer
155 views

Showing a polynomial irreducible

How to show that the polynomial $Y^2+X^2(X-1)^2$ is irreducible in $\mathbb R[X,Y]$. I tried to show that $\mathbb R[X,Y]$ modulo this ideal is an integral domain but I cannot find any homomorphism.
2
votes
1answer
42 views

Question on syzygies

It is hard to formulate a question, but I want to ask about a reference/recipe for computing syzygies in general. For example, on $\mathbb{P}^1_{(x:y)}$ there is an exact sequence $0\longrightarrow ...
2
votes
2answers
37 views

Eccentricity is invariant for ellipse defined by intersection between plane and ellipsoid [can't be correct]??

When a plane intersects a sphere, the intersection is always a circle, due to rotational symmetry etc. However, if a plane intersects an ellipsoid( say, the rotation of ...
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0answers
25 views

What is the category of models?

I'm taking a look at Introduction to Algebraic Geometry and Algebraic Groups by Michael Demazure and Peter Gabriel, and I'm confused about some terminology. Early in the book it says "A ...
1
vote
2answers
43 views

Line in projective space is an example of a curve of genus $0$?

Let $L$ be a line in the projective space $\mathbb{P}^n$ over a field $k$. Is a line $L$ an example of a curve of genus $0$ in $\mathbb{P}^n$. I was wondering if I could verify this with someone, ...
2
votes
0answers
28 views

Cohomologies of certain vector bundle on $\mathbb{P}^3$

Consider the collection of $m$ pairwise disjoint lines $L_1,\ldots,L_m$ in $\mathbb{P}^3$ and pose $Z=L_1\sqcup\cdots\sqcup L_m$. Consider the rank-$2$ vector bundle on $\mathbb{P}^3$ which is given ...
0
votes
0answers
13 views

Intersection Multiplicity of Rational Plane Curves

Suppose I have two rational curves in the complex projective plane. I know their parametrizations, $<x_1(t),y_1(t)>$ and $<x_2(t),y_2(t)>$ I know I can use Grobner bases to find an ...