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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57 views

Projective curve $x^d+y^d+z^d=0$ is nonsingular using Jacobian matrix

According to this question: Nonsingular projective variety of degree $d$, the curve $x^d+y^d+z^d=0$ in $\mathbb{P}^2$ is nonsingular. I'm trying to prove this. Hartshorne defines nonsingular ...
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51 views

On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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36 views

Zariski cohomology of $\mathbb{A}^1$ over a local ring with values in $\mathbb{G}_m$

Let $X$ be a the spectrum of a regular local ring. What is known about the vanishing of the Zariski cohomology group $$ H^n(\mathbb{A}^k_X,\mathbb{G}_m) $$ for $n,k\geq 0$? If $X$ has dimension $d$ ...
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53 views

Direct image of the exceptional divisor along a blow-up

Let $X=\mathrm{Spec}(k[x_1,\ldots,x_n])$ for $n\geq 2$, and let $\mathcal{I}=\widetilde{I}\subseteq\mathcal{O}_X$ for an ideal $I\subseteq k[x_1,\ldots,x_n]$. Let ...
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46 views

If $G/H$ and $G$ are connected linear algebraic groups must $H$ also be connected?

Let $k$ be a perfect field (e.g of characteristic zero) and let $G$ and $H$ be linear algebraic groups over $k$, with $H$ a normal subgroup of $G$. If both $G$ and $G/H$ are connected, must $H$ ...
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26 views

A Small Problem on the Surjectivity of a Map Between Two Locally Free Sheaves

I met a problem when reading Positivity In Algebraic Geometry I. It is Example $1.8.15$ in Chpater 1 Section 8, which is called Green's Theorem. Let $W$ be a subspace of $H^{0}(P, O_{P}(d))$, and $W$ ...
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41 views

differential of $f:X\to\Sigma$ as an elliptic surface,

Let $X$ be an algebraic surface surface and $\sum$ an algebraic curve, and assume, $f:X\to\Sigma$ be an elliptic surface, my question is Why the differential $df$ can be viewed as an injection of ...
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1answer
39 views

finding $\lambda$ when equation of parabola is given

If the equation $\lambda x^2 + 4xy + y^2 + \lambda x + 3y + 2 = 0$ represents a parabola. Then find $\lambda$. I got stuck in this question while solving parabola. Is here anybody who can help me ...
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46 views

Definition of the $\Bbb C^*$-weight of a line bundle

I'm new to geometric invariant theory and am unsure about a definition. Let $X$ be a smooth projective-over-affine variety equipped with a $\Bbb C^*$ action and let $Z$ be the fixed locus of this ...
3
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1answer
101 views

Geometrical description of maps of schemes

In preparation for an exam, I am trying to solve the following question: Describe geometrically all maps from $\operatorname{Spec}(\mathbb{C}[z]/(z^2))$ to $\operatorname{Spec}(\mathbb{C}[x,y])$. ...
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21 views

Graph of a regular function. When is the projection on the first component birational?

Let $X$ be an irreducible variety over a field $k$ and $f$ a regular function on some open subset $U\subseteq X$. Let $F\subseteq X\times \mathbb{A}^1_k$ be the graph of $f$ and suppose that the graph ...
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32 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
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2answers
48 views

Map from $\mathbb{C}^2$ to $\mathbb{A}_{\mathbb{Q}}^2$

I was doing Exercise 3.2 I on Ravi Vakil's notes on Algebraic Geometry: consider the map of sets $\phi: \mathbb{C}^2 \rightarrow \mathbb{A}_{\mathbb{Q}}^2$ defined as follows. $(z_1, z_2)$ is sent to ...
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63 views

Quotient of locally free sheaf is locally free?

If $0\rightarrow F\rightarrow G\rightarrow H \rightarrow 0$ is an extension of $\mathcal{O}$-modules with $F$ and $G$ locally free (each of constant finite rank, i.e. vector bundles), then is $H$ ...
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183 views

Looking for a smooth curve that is not rational

I am preparing for an exam in (mostly classical) algebraic geometry, and I have some preparatory questions, among which: Can you write the equations of any nonsingular curve in any projective ...
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1answer
60 views

Completion of quotient of polynomial ring

Hartshorne's Algebraic Geometry uses the following facts on page 35 without proof: The completion of $(k[x,y]/(y^2-x^2-x^3))_{(x,y)}$ is $k[[x,y]]/(y^2-x^2-x^3)$ and that of ...
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44 views

What does “$\overline{G}_*$ is the residue of $G_*$ in $\mathscr{O}_P(F)$” mean in Fulton's book on algebraic curves?

I'm trying to understand this phrase in Fulton's algebraic curves book page 53: Anyone could help me? Thanks
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1answer
70 views

Can $xy=0$ be the image of an algebraic morphism $\mathbb A^2 \rightarrow \mathbb A^2$?

Suppose we have an algebraic morphism $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$. Can the image of $f$ be the zero locus of the polynomial $xy$? I think not, at least not if we're working over ...
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1answer
32 views

Help in this characterization of the gaps of the symmetric numerical semigroups

Before my question, some background: Definition 1: A numerical semigroup is a subsemigroup $N$ of the additive semigroup $\mathbb N$ of the non-negative integers such that $\mathbb N-N$ is ...
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195 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
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1answer
147 views

$k$-algebra homomorphism of the polynomial ring $k[x_1,\dots,x_n]$

Let $\phi:k[x_1,\dots,x_n]\mapsto k[x_1,\dots,x_n]$ be a $k$-algebra homomorphism with $\phi(x_i)=f_i$, where $k$ is algebraically closed and has characteristic zero. I have the following questions: ...
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35 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
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1answer
47 views

Scheme theoretic dual of $\mathbb P^n_k$

Consider an algebraically closed field $k$, and define $\mathbb P^n_k:=\textrm{Proj}(k[T_0,\ldots,T_n])$. In some algebraic geometry books I see the notation ${(\mathbb P^n_k)}^\vee$ that is referred ...
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1answer
85 views

Flatness after dividing out a minimal prime ideal

Let $A \hookrightarrow B$ be an extension of finitely generated, reduced $k$-algebras, where $k$ is a field of characteristic zero such that $B$ is a free $A$-module of finite rank. Let $A$ be an ...
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46 views

Hartshorne Exercise III 6.2 (a)

Let $X=\mathbb{P}^1_k$, with $k$ an infinite field. Show there does not exist a projective object $\mathcal{P}\to\mathcal{O}_X\to 0$. The author suggests to consider surjections of the form ...
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1answer
119 views

Meaning of notation $\operatorname{ord}_Q(g)$ in “Algebraic Curves” by Fulton

I didn't understand this notation in the chapter 7 page 93 of Fulton's algebraic curves book: What the author means by $\text{ord}_Q(g)$? Maybe he would like to say $\text{ord}_Q(G) := ...
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1answer
55 views

Help in this equivalence in Fulton's book

I'm studying Fulton's algebraic curves book and some of its notation look like confusing. I need help to understand this equivalence in the page 35. The preceding paragraph he mentioned I put ...
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32 views

Property that defines Quadric Surface

The book < Geometry and the Imagination > (written by David Hilbert) introduces a property of a Quadric Surface without a proof. Property : The cone consisting of all the tangents from a ...
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1answer
43 views

Number of roots of two polynomials

Let $f,g\in R[X,Y]$ and suppose that $\{ (x,y) \in R^2 : f(x,y)=g(x,y)=0\}$ is finite. If the variable $X$ occurs only up to degree $d_1$ in both $g,f$ and $Y$ only up to degree $d_2$ is there ...
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25 views

Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Especifaclly in the context of iteratation of holomorphics functions. This concept could be taken to some of the deformation ...
2
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1answer
95 views

Reference request: Some theorems in an article of Grothendieck.

In "Standard conjectures on algebraic cycles" Grothendieck says: "The first is an existence assertion for algebraic cycles (considerably weaker than the Tate conjectures), and is inspired by and ...
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1answer
49 views

The intuition behind the coordinate ring $\Gamma(F)$

I'm studying Fulton's algebraic curves book. He gives the following definitions: We can define the coordinate ring of a nonempty variety $V\subset \mathbb A^n$ as $\Gamma(V)=k[X_1,\ldots,X_n]/I(V)$. ...
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47 views

The Kähler form and the anticanonical line bundle

Let $M$ be a Kähler manifold. We say that $M$ is Fano if the anticanonical line bundle $K_M^*$ of $M$ is ample (or positive). On the other hand, I sometimes see the following definition (or ...
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1answer
48 views

Maximal homogeneous ideals of a graded $k$-algebra.

Let $k$ be an algebraically closed field, and let $A$ be a finitely generated commutative $k$-algebra. Given any maximal ideal $\mathfrak{m}\subset A$, we can form the quotient to obtain a map $A\to ...
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2answers
65 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
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17 views

Reference request: About some important result in a book of Lefschetz.

Is there a (modern)book in which the most important results of L'analysis situs et la géométrie algébrique, Lefschetz" are exposed?
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228 views

Another Algebraic de Rham Cohomology question…

NOTE: scroll down to read my latest edit first if you're reading this for the first time :) My aim is to calculate the de Rham cohomology of the variety $U = \text{Spec} \ A$, where: $$A = ...
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1answer
21 views

Why this $F_*=F(X,Y,1)$

I'm studying Fulton's algebraic curves book. Someone could help me to prove this phrase highlighted: I didn't understand why the $F_*$ he defined is the same of the known $F_*=F(X,Y,1)$. Thanks ...
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1answer
58 views

Ideal of affine piece of a projective variety

The main question I want to eventually wind up answer is part b of Corollary 2.6 in Chapter 1 of Silverman's "The Arithmetic of Elliptic Curves". The proposition is Let $V$ be a projective variety. ...
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128 views

Standard spaces with non-standard topology and around

The following series of questions gives me no rest. Let $\mathbb{\widetilde{R}}^n$ be $\mathbb{R}^n$ with Zariski topology, i.e. we say that $A\subset \mathbb{R}^n$ is closed if it is given by the ...
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2answers
42 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
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1answer
36 views

Morphism schemes after base extension

I have been reading Ravi Vakil notes on algebraic geometry, and one exercise asks if $phi:X\rightarrow Y$ and $\pi:X\rightarrow Y$ are morphisms of $k$-schemes and $\ell/k$ a field extension, if ...
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Action on Picard group of a blown up surface.

In this paper on page 45 a birational mapping $\psi: \mathbb P\times\mathbb P \rightarrow\mathbb P\times\mathbb P$ is defined via $\psi(f,g) = (\frac{f-g}{2-f-g},\frac{f-g}{2+f+g})$. Blowing up gives ...
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3answers
92 views

The topology on $\mathbb A^2$ is not the product topology [duplicate]

I'm trying to prove the Zariski topology on $\mathbb A^2$ is not the product topology on $\mathbb A^1\times \mathbb A^1$. I'm looking for a counter-example based on the fact the closed subsets in ...
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30 views

how to lift geometrically integralness using etale(+something else) morphisms

Take $X$ and $Y$ to be $k$-varieties, where $k$ is a field of characteristic 0. Assume also that $X$ is geometrically integral, and let $f: Y \to X$ be an etale a morphism of $k$-varieties. Question: ...
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33 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
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2answers
90 views

defining the group law on elliptic curves in general

Let $k$ be an arbitrary field and $C \subset \mathbb{P}^2(k)$ an elliptic curve. In order to define the group law on $C$ we need to establish some geometric facts first, e.g. Any line intersects $C$ ...
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223 views

Sard's theorem for algebraic varieties

(One version of) Sard's theorem states that: Theorem (Sard): Given $M$ and $N$ smooth manifolds of dimensions $m$ and $n$ respectively, and a smooth map $f:M\to N$, then the set of singular values ...
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38 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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1answer
32 views

Quantum Cohomology of Affine Toric Varieties

I would like to know whether quantum cohomology rings of affine toric varieties have been calculated, if this is possible. Does anyone have a relevant paper they could refer me to? I have seen it ...