6
votes
2answers
117 views

Push forward of the structure sheaf along covering

Let $f: X \to Y$ be a covering (proper, surjective, finite regular map) of smooth projective varieties of degree $d$. How one can show that in this case $f_* \mathcal{O}_X$ is a locally free sheaf of ...
2
votes
1answer
93 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 ...
4
votes
0answers
42 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 ...
1
vote
0answers
16 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?
0
votes
1answer
69 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 $C$ ...
1
vote
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 ...
1
vote
0answers
49 views

Grothendieck-Serre duality for complete smooth varieties

Let me fix a complete (but not necessarily projective) smooth variety $X$ over an algebraically closed field $k$ of characteristic $0$. Denote $d=\operatorname{dim}X$, and $\omega=\Omega^d_X$. I'd ...
5
votes
1answer
84 views

What are the prerequisites for Fulton's “Intersection Theory”?

Is it necessary to read SGA VI to understand "Intersection Theory" by William Fulton?
3
votes
0answers
34 views

Where can I find some articles of Weil.

Where can I find the articles of Weil: Variétés abéliennes et courbes algébriques Sur les courbes algébriques et les variétés qui s'en déduisent. on Internet?
2
votes
1answer
94 views

Bezout's bound and resultants - reference request

In Terry Tao's blog post about Bezout's inequality, he writes: In our notation*, this theorem states the following: Theorem 1 (Bezout’s theorem) Let $d=m=2$. If $V$ is finite, then it has ...
-1
votes
1answer
29 views

Reference request: About Weil book

In "Standard conjectures on algebraic cycles" of Grothendieck and "Algebraic cycles and the Weil conjectures" of Kleiman they say in their references: A. Weil: Variétés Kählériennes, Hermann, ...
7
votes
3answers
302 views

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? I'd like the lectures to cover main topics of the books ...
1
vote
0answers
24 views

Where to learn about the Chow scheme and the Hilbert-Chow morphism?

I would like to learn something about the Chow scheme of cycles on an algebraic variety. I am not after an abstract treatment of the moduli problem in full generality, actually I would be happy with a ...
1
vote
0answers
36 views

Reference request on numeric semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
1
vote
0answers
29 views

Quotients of varieties by polynomial relations

Let $V$ be an affine variety in $\mathbb{C}^{n}$, i.e. $V$ is the vanishing set of an ideal $I \subset \mathbb{C}[x_{1}, \dots, x_{n}]$. Furthermore let $g \in \mathbb{C}[x_{1}, \dots, x_{n}]$. ...
1
vote
0answers
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 ...
2
votes
0answers
51 views

Quotient of smooth variety is smooth if fixed point set is a divisor?

I've heard (a variant of) the following result being mentioned , but haven't been able to find a reference. I would like to know if the following is true, and if so, I'd very much appreciate a good ...
2
votes
2answers
195 views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
2
votes
1answer
93 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
votes
1answer
126 views

Blow-ups in Projective Space

This is in regards to a question (no solutions or comments thus far :-() I asked earlier in regards to the blow-up of an elliptic curve: Question Let $f(x,y)=y^2-4x^3+ax+b$, where $(x,y)\in\mathbb ...
2
votes
1answer
73 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
2
votes
1answer
187 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
1
vote
1answer
69 views

Beggining in Algebraic Geometry

My question is about sources for start the study of algebraic geometry. I know that it requieres so much algebra, but, is there any book which can be readed without many tolos of modules, Galois, ...
6
votes
1answer
81 views

Affine varieties, the Brauer group and gerbes

For an affine variety $V$, the Brauer group $Br(V)$ and the cohomological Brauer group $Br'(V)$ (i.e. the torsion subgroup of $H^2_{et}(V,\mathbb{G}_m)$) coincide, by results of various authors ...
3
votes
1answer
90 views

Learning roadmap for classical algebraic geometry (italian school)

Can someone suggest a learning roadmap for classical algebraic geometry as develop by the great italian school? Severi has a few books but in italian. I would like to know what are the best English ...
3
votes
0answers
89 views

Developing intuition in algebraic geometry through differential geometry?

I'm interested in algebraic geometry (I am working through Ravi Vakil's notes and also have worked with curves and general varieties in the past), and have seen some basic definitions from ...
2
votes
1answer
80 views

Explicit Kähler forms and Kähler cone of one-point blowup of $\mathbb{CP}^2$

I am interested in understanding the Kähler cone of the one-point blowup of $\mathbb{CP}^2$, also known as the first Hirzebruch surface. Let's call this manifold $\Sigma_1$, and call its Kähler cone ...
1
vote
1answer
33 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
3
votes
0answers
56 views

$h^{p,q}$ of projective space

How can we calculate the Hodge number $h^{p,q}= \dim H^p(\mathbb{P^n},\Omega^q_{\mathbb{P}^n})$ of projective space? Is there a reference for that?
2
votes
0answers
80 views

Books on algebraic surfaces

I am interested in learning about algebraic surfaces (e.g. their classification in characteristic 0), and I was wondering whether any knowledgeable people would be so kind as to give their thoughts ...
2
votes
1answer
119 views

What should a student (with algebraic-geometry minded) study in differential geometry?

One of my friend who is an undergraduate student, has known something about algebraic geometry (equivalent to chapter 1 and a little bit chapter 2 in GTM 52 by Hartshorne). He is now has to study a ...
3
votes
1answer
89 views

Resolution of Singularity

Consider $y^8=x^{11}$. So, $y^8-x^{11}=0$. Define $f(x,y)=y^8-x^{11}$. Then, $\nabla f = (f_x,f_y)=(-11x^{10},8y^7)$. For $(f_x,f_y)=(0,0)$, we must have $(x,y)=(0,0)$; the singular point. ...
6
votes
4answers
213 views

Learning about Grothendieck's Galois Theory.

I have background in category theory and I am familiar with the very basics of algebraic geometry - Chapters I and II of Hartshorne. What would be a recommended (self-contained, maybe?) text for ...
1
vote
0answers
45 views

A question on the notation in Hartshorne's Algebraic Geometry

My confusion is in III.Corollary $9.4$. By $\mathcal{F}_y$ does he mean the pull-back of $\mathcal{F}$ by the closed immersion of $X_y$ into $X$? By $\mathcal{F} \otimes k(y)$ does he mean the ...
0
votes
1answer
45 views

surjective morphism of varieties

Let $A,B$ be two varieties (integral finite type separated scheme) over a field $k$ and let $f : A \to B$ be a proper morphism. Is it true that $f$ is surjective if and only if $dim(\overline{f(A)}) ...
2
votes
2answers
122 views

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big?

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ? Is there a formula relating the dimension of the Zariski tangent space and the order of ...
5
votes
0answers
187 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
7
votes
2answers
111 views

Techniques for showing an ideal in $k[x_1,\ldots,x_n]$ is prime

An affine variety $X$ over a field $k$ is irreducible if and only if its defining ideal $I(X)$ is prime (in this post we use the convention that varieties are not necessarily irreducible). Hence, it ...
3
votes
0answers
47 views

Can the Milnor number be used to resolve curve singularities?

Let $f(x,y)\in \mathbb{C}[x,y]$ define a curve $C$ which is singular at the origin. By successively blowing-up the origin, we can resolve the singularities of $C$. Of course to make sure this process ...
1
vote
1answer
41 views

Parabolic Cusp of an Action on the Upper Half Plane

This is a basic definition question. Parabolic bundles are used in certain counting arguments in my research area. I asked my advisor for a reference on these, and he directed me to the paper of Mehta ...
2
votes
1answer
108 views

Mumford-Oda - Algebraic Geometry II . There will be a complete book?

Online there is the draft of a book written by Mumford and Oda that should be the continuation of "Algebraix Geometry I complex projetive varieties" (Mumford,1976). Do you know if and when this book ...
1
vote
0answers
103 views

Note or book on Examples of regular, Gorenstein, Cohen Macaulay, … rings

I need a good note or book with plenty of examples in commutative algebra and algebraic geometry which surveyed being regular, Gorenstein, Cohen Macaulay, .... Can you help? thanks.
3
votes
0answers
112 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
2
votes
0answers
45 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
1
vote
1answer
101 views

Book Recommend Differential Geometry of Algebraic Manifolds

I just want to study Differential Geometry of Algebraic Manifolds. but I can`t find a book about that. Is there any good book for studying Differential Geometry of Algebraic Manifolds??
14
votes
1answer
198 views

Global Optimization and Real Algebraic Geometry

Wikipedia suggests that: "Methods based on real algebraic geometry" are some of the "most successful general strategies" for solving global optimization problems. Could someone suggest an reference ...
5
votes
1answer
84 views

Which pullbacks preserve the sheaf of differentials?

Let $f : X \to Y$ be a morphism of $S$-schemes. Assume that the canonical morphism $\alpha : f^* \Omega^1_{Y/S} \to \Omega^1_{X/S}$ is an isomorphism. What can we say about $f$? Is $f$ formally étale? ...
1
vote
0answers
79 views

Coherent sheaves of finite length

Where can I find a treatment of coherent sheaves of finite length over, say, Noetherian schemes? Just things as basic as their definition and elementary facts about them. I am familiar with modules of ...
2
votes
1answer
55 views

Orbits that 'coalesce'

Let $R$ be a commutative ring, $G$ a group scheme over $\mathrm{Spec}\;R$, and $X$ a scheme over $\mathrm{Spec}\;R$ on which $G$ acts $R$-morphically via $G\times X\to X$. Suppose $S$ is another ...
5
votes
0answers
57 views

If $U\to X$ is a closed immersion, $U$ is dense in $X$ and $X$ is reduced, why is the closed immersion an isomorphism?

This came up in the Reduced-to-separated theorem. If $U\to X$ is a closed immersion, $U$ is dense in $X$ and $X$ is reduced, why is the closed immersion an isomorphism?