3
votes
0answers
34 views

Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces. Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex ...
1
vote
0answers
27 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
2
votes
0answers
21 views

Base change is exact for algebraic groups

I need a reference for the following fact: let $1 \to G' \to G \to G'' \to 1$ be a ses of algebraic groups over $S$. Let $S' \to S$ be a base change. Then $1 \to G'_{S'} \to G_{S'} \to G''_{S'} \to ...
1
vote
0answers
33 views

Cohomology and Base Change - Degree 0 Sanity Check

Both Vakil and Hartshorne describe Cohomology and Base Change in the following way: Suppose $f:X \rightarrow Y$ is a projective (in Vakil, proper) morphism of Noetherian schemes, $F$ a coherent ...
1
vote
1answer
37 views

Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic?

Let $f: X \to Y$ be a morphism of schemes. It produces a pair of adjoint functors $f^*$ and $f_*$ on the category of quasi-coherent sheaves i.e. there is a natural isomorpism $$ ...
3
votes
0answers
46 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...
5
votes
0answers
42 views

How can a finite graph be viewed as a discrete analogue of a Riemann surface?

In the paper "Riemann–Roch and Abel–Jacobi theory on a finite graph" by Baker and Norine, the first line of the abstract states: "It is well known that a finite graph can be viewed, in many respects, ...
1
vote
0answers
27 views

EGA reference for equivalent criteria for ampleness

Let $X$ be a projective scheme over a field $k$ with $\mathcal{L}$ a very ample line bundle on $X$ (very ample here means relative to the structure morphism $X \to \operatorname{Spec} k$. Where is it ...
4
votes
0answers
54 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
2
votes
0answers
38 views

Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
1
vote
0answers
53 views

Does such a polynomial map always exist?

First question: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is ...
2
votes
1answer
59 views

What is the importance of Jacobian Conjecture and any progress on it?

What is the importance of Jacobian Conjecture?Are there any important central problem with the conjecture as precondition? and any progress on it?
1
vote
1answer
68 views

Analog of holomorphic Lefschetz fixed point theorem for smooth algebraic varieties

If $X$ is a compact complex manifold and $f: X \to X$ is a holomorphic map with isolated nondegenerate zeroes. Then there is a version of Lefschetz fixed point formula with traces on Dolbeaut ...
0
votes
0answers
37 views

Can regular functions be specified simply as the sections of a bundle?

(following on from this question of mine). In Hartshorne Algebraic Geometry the definition of the sheaf of rings attached to the spectrum of a ring $A$ to define an affine scheme has the following: ...
2
votes
2answers
53 views

Reference for Algebraic Groups in Ergodic Theory

It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that assumes a knowledge of commutative algebra ...
3
votes
1answer
52 views

Where in EGA is the result of Serre Vanishing located?

We recall Serre's Vanishing Theorem which states the following. Let $X$ be a closed subscheme of $\Bbb{P}^n_A$ with $A$ a Noetherian ring. Then for any $\mathcal{F} \in \operatorname{Coh}(X)$, there ...
4
votes
2answers
85 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
6
votes
2answers
168 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
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 ...
4
votes
0answers
49 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
18 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
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 $C$ ...
1
vote
1answer
37 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
55 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
101 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
35 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
105 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
31 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, ...
8
votes
3answers
355 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
33 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
37 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
30 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
45 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
57 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
239 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
142 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 ...
3
votes
1answer
79 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
197 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
73 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
85 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
104 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
99 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
99 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
35 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
58 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
84 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
126 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
91 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
238 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 ...