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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Vector Bundles:differential geometry vs algebraic geometry

I am in trouble about the vector bundle part in the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle, I know what is a vector bundle(or a fibre bundle) in the differential geometry, ...
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11 views

Finding a bijective morphism

I am given two Varieties $Z=V(x^2+y^2+1) \subset C^2$ and $W=V(x^2-y^2-1) \subset C^2$. We need to find a bijective morphism f such is an isomorphism with the inverse of f. First how we defined ...
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11 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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25 views

On the proof of Lemma 3.17 in Harris - Algebraic Geometry, A First Course

This question is motivated by the proof of Lemma 3.17 in Algebraic Geometry, First Course by Joe Harris. Let $Y$ be a closed subset of $\mathbb{A}^n$ and consider the projection $\pi : Y \times ...
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1answer
23 views

The genus of a certain kind of cubic

I have a cubic curve that looks like $$ a_0 x^3 + a_1 x^2 y + a_2 xy^2 + a_3 y^3 = b $$ with $a_0, a_1, a_2, a_3$, and $b$ all integers, and $a_0$ and $b$ nonzero. I'm not sure but I think in my ...
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19 views

How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...
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1answer
19 views

Commutative Algebra and Monomial orders

So whenever we are doing any problem related to ideals in the polynomial ring $k[x_{1},x_{2},\dots x_{n}]$,(e.g. calculating a grobner basis for instance or doing the division algorithm for a set of ...
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22 views

About rational curves on elliptic K3 surfaces

I am trying to read this paper http://arxiv.org/pdf/math/9902092.pdf. I have some trouble at the end in Corollary 3.27, and also Proposition 3.24. 0)Morally speaking my main trouble is that i don't ...
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35 views

Projective line intersecting 3 projective subspaces

I am trying to solve the following problem: Let $\mathbb{P}(U),$ $\mathbb{P}(V)$ and $\mathbb{P}(W)$ be projective subspaces of dimension $k,$ $l$ and $m$ respectively in $\mathbb{P}_K^n$. Suppose ...
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0answers
19 views

Singular Conics and Intersection of Line with a Conic

I've been working through Silverman and Tate's book Rational Points on Elliptic Curves. They use conic equations as an introduction to singular/nonsingular curves. I've reproduced the problem with my ...
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1answer
24 views

WLOG doubt: why can we assume that two disjoint linear subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$ are given by the following equations…

Let $H_1,H_2$ be two linear disjoint subspaces of dimension $n$ in $\mathbb{P}^{2n+1}$. Let $(x_0:\cdots:x_n:y_0:\cdots:y_n)$ be homogeneous coordinates in $\mathbb{P}^{2n+1}$. My question is: ...
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1answer
16 views

Crushing Cylindrical wooden pillar

The Crushing load of a cylindrical wooden pillar varies directly as the fourth power of its diameter and inversely as the square of its height. If K is the crushing load of a certain wooden pillar, ...
3
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1answer
39 views

Tangent lines of a smooth curve $C \subseteq \mathbb{P}^2$

Let $C$ be a smooth curve, given as the zero locus of a homogeneous polynomial $f \in \mathbb{K}[x_0,x_1,x_2]$. Consider the morphism $\varphi_C:C\rightarrow\mathbb{P}^2$ such that ...
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0answers
19 views

Question about bilinear and quadratics form

I'm reading this book: Geometry of algebraic curves by Cornalba, Harris etc. At page 289 there is an excercise where the authors define a quadratic form $Q:V \times V \rightarrow \mathbb{C}$ taking ...
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1answer
17 views

Subgroups of points of order 2 in an elliptic curve

Depending on the roots of $y^2 - x(x^2+ax+b) = 0$ being real or not, we can have 2 subgroups of points of order 2 for a given elliptic curve- Kelin-4 group or a cyclic group of order 2. How does one ...
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47 views

A computation related to Hironaka's Example

My questions are at the very end, first I'll describe the context. Let $f:\mathbb{P}^3\to \mathbb{P}^3$ be an involution whose fixed locus consists of two disjoint lines $L, L' \subset \mathbb{P}^3$. ...
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38 views

Blowup of six points in $\mathbb{P}^2$

I have been looking at sequential blowups of six points $P_i$ in $\mathbb{P}^2$. In the general case, we can identify the blowup with a nonsingular cubic surface in $\mathbb{P}^3$ and under this ...
3
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28 views

Intersection points of two Bernoulli lemniscates

What is the maximum number of intersection points of two Bernoulli lemniscates in the real plane? Here is some of my efforts: A Bernoulli lemniscate is a degree four curve with two nodes on the ...
2
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1answer
34 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...
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36 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
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1answer
39 views

Why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$?

I have a very brief question. If you have a bunch of line bundles $L_1,\dots,L_p$ over a scheme $S$, why does $\bigwedge^p(L_1\oplus\cdots\oplus L_p)\cong L_1\otimes\cdots\otimes L_p$, and can't find ...
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37 views

Algebraic closedness in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...
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2answers
119 views

How is the notion of adjunction of two functors usefull?

Is there a secret or an intuitive idea behind the fact of creating the concept of adjunction of two functors ( Functor - Adjoint Functor ) ? How is this notion of adjunction of two functors usefull ? ...
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43 views

Tangent Cone of a Complete Intersection

Can you give me an example of an affine variety $X \subseteq \mathbb{A}^n_{\mathbb{C}}$ over the complex numbers which is a complete intersection such that the reduced tangent cone at some point $p ...
5
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1answer
37 views

Write $\mathbb{P}^3_{\mathbb{C}}$ as a union of disjoint lines

Is there a set $\Gamma=\{L \subseteq \mathbb{P}^3_{\mathbb{C}}: L \textrm{ is a projective line}\}$ such that every point $p \in \mathbb{P}^3_{\mathbb{C}}$ lies on exactly one line $L_p \in \Gamma$? ...
2
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0answers
33 views

Union of affine varieities is a projective variety?

Let $X \subset \mathbb{P}^n$ be a subset and let $U_i = \{ [z_0: \cdots :z_n] : z_i \neq 0 \}$ for $0 \leq i \leq n$ be the usual affine cover of projective space. Suppose that $X \cap U_i$ is an ...
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2answers
77 views

Cohen-Macaulay rings and Normal rings

is there an example that R is Cohen-Macaulay but not normal ring? what about the converse example?
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1answer
22 views

What is vectors straddle a plane mean?

There is a condition in a paper, saying that two vectors straddle a plane. How can we transfer this condition to a equation? Because I have another 5 equations and need this one to solve 6 unknowns. ...
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1answer
52 views

Why is this map injective?

On the page we find the following: $ \phi : Z ( P_1 , \dots , P_r ) \to \mathrm {Spm} (K[ X_1 , \dots , k_n ] / \sqrt{ ( P_1 , \dots , P_r )}) $ defined by $ \phi ( ( a_1 , \dots , a_n ) = \pi ( ( ...
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1answer
33 views

Number of inflection points of an algebraic projective curve

I' m trying to prove that a curve in $\mathbb{P^{2}(\mathbb{C})}$ of degree $d$ has an infinite number of inflection points or it has at most $\le 3d(d-2)$ inflection points. Let be $C$ the curve and ...
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19 views

Reflexive sheaves on stable curves-II

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...
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1answer
23 views

Reflexive sheaf on normal surfaces

Let $X$ be a normal, projective scheme of pure dimension $2$ and $\mathcal{F}$ is a reflexive coherent sheaf on $X$. Is $\mathcal{F}$ locally free?
2
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1answer
42 views

Blow-up toric varieties.

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory ...
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1answer
15 views

How does an irreducible quadric in projective space look like?

I read the answer to the following question: Quadrics are birational to projective space Here it is stated that: Over a field $k$ of characteristic ≠2 every irreducible quadric $Q \subset \mathbf ...
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2answers
39 views

Jacobian of n linearly independent forms in n variables

Let $k$ be a field of characteristic zero and let $f_1, \ldots, f_n \in k[x_1, \ldots, x_n]_d$ be linearly independent forms of degree $d$ in $n$ variables. Is there a nice algebraic argument for ...
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0answers
25 views

How to understand the elementary modification?

In the Friedman's book: Algebraic Surfaces and Holomorphic Vector Bundle , there is concept elementary modification at Ch2 Def15. Let V is rank 2 bundle on X and L a line bundle on effective divisor ...
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0answers
13 views

Automorphisms of del Pezzo surface of degree $1$.

I have som problems with understanding of finite subgroups $G$ of $Aut(S)$,where $S$ del Pezzo surface of degree $1$. I consider case, when $k = \mathbb{Q}$. I don't understand why $Aut(S)$ embedding ...
2
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0answers
23 views

Lifting of Commuting Maps of Vector Bundles

Assume that we have a vector bundle $\mathcal{F}$ over $\mathbb{P}^d(\mathbb{C})$ that is generated by global sections. Let $\pi \colon \mathcal{O}^n \to \mathcal{F}$ be the associated map that is ...
5
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0answers
57 views

Does $L_1\oplus\mathbb{A}^1_X\cong L_2\oplus\mathbb{A}^1_X$ imply that $L_1\cong L_2$?

Suppose $L_1,L_2$ are line bundles over a scheme $X$. If one knows that $L_1\oplus\mathbb{A}^1_X$ and $L_2\oplus\mathbb{A}^1_X$ are isomorhpic, is that enough to conclude that $L_1$ and $L_2$ are ...
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1answer
41 views

Intersection of a hypersurface with a projective variety [on hold]

I don't understand the argument in the proof of Corollary 3.15 (This is from Harris). In particular, how exactly is Corollary 3.14 applied?
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1answer
36 views

Sections of Divisors on Projective Space

Everything is over $\Bbb{C}$. Let $X$ be a smooth projective variety. Fix an open covering $U_i$ of $X$ and let $D$ be a Cartier divisor given by a collection of rational functions ...
3
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1answer
26 views

Are points in general position generic points?

In Harris' algebraic geometry book, $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are said to be in general position if no $n+1$ or fewer of them are dependent. I want to prove that, if ...
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14 views

Quick question: Pull back under double cover of tangent space on the projective plane is stable?

Let $f:\mathbb{P}^1\times\mathbb{P}^1\rightarrow\mathbb{P}^2$ be the double cover branched along some conic $C\subset \mathbb{P}^2$. Is $f^*T_{\mathbb{P}^2}(-1)$ $\mu$-stable/semistable? Is there any ...
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29 views

How do we get this quotient $\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of vector bundles on a surface. Here $M$ and $N$ are line bundles, and so rank $ E$=2. Also, if ...
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1answer
23 views

Map between free sheaf modules not arising from a “matrix”

My question is about this example in the Stacks project. For convenience and completeness, here is the relevant part. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$ of the real line ...
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2answers
42 views

How do I find an isomorphism between varieties

Our book defines an isomorphism between varieties when there exist two maps say $\phi: V \rightarrow W$ and $\psi: W \rightarrow V$ both morphisms and $\psi \circ \phi =id_V$ and $\phi \circ \psi =id ...
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2answers
65 views

What does a polynomial look like under projection of underlying space?

Consider a multivariate polynomial in $F:\Bbb R^3\rightarrow\Bbb R$, $F\in\Bbb R[x,y,z]$ with prescribed values over a sphere in $\Bbb R^3$. Consider standard Riemann projection from $\Bbb ...
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26 views
+50

Global functions functor for derived stacks

On page 24, 25 of the paper Loop Spaces and Connections the authors refer to a functor $\mathcal{O}: DSt_k \rightarrow DGA_k^{op}$ from derived stacks to dg algebras over $k$. It is defined as ...
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1answer
14 views

Correspondence between morphism and ring of regular functions

in Hartshorne it is explained that an morphism of varieties $\varphi:X \to Y$ gives rise to $k$-algebra-homomorphism of $O(Y) \to O(X)$. Now I know by the defining property of morphism that a morphism ...
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1answer
19 views

Every variety contains open affine normal subvariety

How to prove this? I think that the starting point here is to use the fact that the set $\{x\in X \,|\, X\, \text{is normal at}\, x\}$ is open. What do I do next? Thank you in advance.