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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155 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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1answer
92 views

Sheafification: Show that $\tilde{\mathscr{F}_x}=\mathscr{F}_x$.

My today's question is about a proof of this book. More precisely we are talking about the proof of Prop. 2.24 on page 52. The book says that we have $\tilde{\mathscr{F}_x}=\mathscr{F}_x$ for all ...
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28 views

Fundamental group of family of rational varieties

Let $X$ be a smooth, projective, simply connected variety over a field $k$ (i.e. $\pi_1^{\text{et}} = 1$). Let $f: Y \to X$ be a family of rational varieties parametrized by $X$, such that $Y$ is ...
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88 views

Can Lagrange multipliers be used to give a good bound on the number of critical points?

I will explain my problem by illustrating a simple case. Easy question: Let $f(x,y)$ be a "generic" polynomial in two variables, of total degree $\le D$. What's a good upper bound for how many ...
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190 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 ...
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42 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
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41 views

Differentials on a curve

Say I have an algebraic curve $C$ over a field $k$ and a group $G$ acting on $C$. Under what conditions on $C$ and/or the action of $G$ on $C$ can one conclude that $H^0(C,\Omega^1_C)^G = ...
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53 views

Covering a closed disk in a rigid analytic space by residue classes

Recently I have been reading through the PhD thesis of Dr. Louis Brewis, "Ramification theory of the p-adic open disc and the lifting problem", which is available free here: ...
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61 views

Why does $H^1(X,\mathbb{Z})$ span a full lattice in $H^1(X,O_X)$ if $X$ is kahler?

We have the exponential short exact sequence for compact complex manifolds. Why does the image of $H^1(X,\mathbb{Z})$ span $H^1(X,O_X)$ over $\mathbb{R}$ if $X$ is kahler? The map is injective, I was ...
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86 views

Application of Zariski's Main Theorem

Suppose $f: A \to B$ is a local homomorphism, $B$ is isomorphic to a localization of an $A$-algebra of finite type. Let $L$ be the field of fractions of $B$, and suppose that $B$ contains the normal ...
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60 views

Pullback of simple normal crossing divisors

It is claim in the book "Higer-Dimensional Algebraic Geometry" by Olivier Debarre (before Theorem 7.21 page 183) that If $D$ is a divisor with simple normal crossing on $X$ and $\pi: Y \to X$ ...
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60 views

Computing localization explicitly for schemes (with a particular view in exercise III 4.7 from Hartshorne)

I'm having some trouble in computing the Čech cohomology for simple schemes (a smooth projective plane curve, for instance). In general, the procedure, apparently, is to find a good cover by affine ...
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34 views

Isomorphism of the etale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$ of complex algebraic varieties. It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(Y)$ is an isomorphism? I think ...
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54 views

Isomorphism of the fundamental groups

Given two complex algebraic varieties $X$ and $Y$. If there exists a birational proper morphism $f\colon X\rightarrow Y$ then a Theorem of Grothendieck (SGA.X) say that $\pi_1^{et}(X)\cong ...
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66 views

Soft: Why does the existence of a singularity cause problems for deRham cohmology?

I've heard that if a variety has a singularity then the deRham theory has "problems". What exactly are these? Im guessing there is some sort of issue with the defintion of a differential form, but ...
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78 views

Fermat Quartic Tiling

I have been reading about the Fermat quartic $Q \subset \mathbb{P}^{2}$, defined in homogeneous coordinates as $X^{4}+Y^{4}+Z^{4}=0$. This is the second most symmetric non-hyperelliptic surface of ...
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66 views

Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
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105 views

Incidence variety fo Grassmmanians

Let $k$ be an algebraic closed field (say, $\text{char}(k)\neq 2$), $n \in \mathbb N\setminus \{0\}$ and $G(m, n) = G(m, \mathbb P^n(k))$ the variety of Grassmmanian of $m$-dimensional linear ...
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1answer
294 views

arithmetic and geometric genus for a reducible plane curve

If $C$ is an irreducible plane curve we have the well known formula relating the airthmetic (obtained via the degree-genus formula) and the geometric genus $$\frac{(d-1)(d-2)}{2} - \sum ...
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65 views

Semistable vector bundles elliptic curve

Let $n=(r,d)$, r=r'n, d=d'n and $M(r,d)$ the moduli space of $S-$equivalence classes of semistable bundles of rank $r$ and degree $d$. How can I construct a finite morphism $M(r',d')^n\to M(r,d)$ ...
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70 views

On the bidegree of a curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence. I'm ...
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51 views

How to find equations that define the image of an algebraic morphism?

Suppose we have a map $f:\mathbb{P}^n\rightarrow \mathbb{P}^m$ which is algebraic. What are the techniques to find the equations defining the image of f as a subvariety of $\mathbb{P}^m$? For example ...
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63 views

2-forms represented by a first Chern class?

Let $M$ be a complex manifold and $\omega$ be a 2-form on $M$. Is there a good way to see whether $\omega$ is represented by the first Chern class of a line bundle on $M$? In other words, when is it ...
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63 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
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58 views

About the calculation of cohomology groups on projective space

In Chapter III, section $5$ of Hartshorne, we want to calculate the cohomology groups of sheaves $\mathcal{O}_X(n)$ on a projective space $X=\mathbb{P}_A^r$, where $A$ is a noetherian ring. We are ...
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170 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.
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69 views

Quadric question

I'm trying to prove that given 3 disjoint lines in $\mathbb{P}^{3}$ there exists a non-singular quadric containing them. The exercise is from the following link: ...
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1answer
92 views

Locally complete intersection in a fiber

Let Y be an affine noetherian scheme, $Z = V_+(F_1, \ldots, F_r)$ a closed subscheme of $\mathbb{P}^d_Y$ that is flat over Y. Let $y_0 \in Y$ be a point such that $Z_{y_0}$ is a complete intersection ...
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113 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
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118 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
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1answer
79 views

Linearization of a group action: why the map is equivariant?

I'm using Dolgachev's book on invariant theory to learn linearizations of group actions. Here is a sketch of main construction: let linear algebraic group $G$ act on a quasi-projective variety $X$, ...
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226 views

Computing toric ideals via saturation

I have recently got interested in toric varieties and I have a question concerning their ideals. Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u ...
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85 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...
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56 views

Extension of morphism of Coherent sheaves over the projective space

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates ...
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62 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
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110 views

no morphism between curves of different genus

Let $C_1,C_2$ be two smooth irreducible curves of genus 4,3 resp. Prove there is no morphism $\phi: C_1 \to C_2$. Well, the tool of treating genera is Hurwitz's theorem, which says here that ...
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1answer
134 views

Book with color pictures of algebraic surfaces

I have a pretty specific question: I'm looking for a book with color pictures of algebraic surfaces. Could anyone point me in the right direction?
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149 views

Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto ...
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188 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
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2answers
106 views

Krull-Schmidt-Remak for vector bundles

I'm reading Nori's paper The fundamental group scheme, and I have some problems in certain passages of the proofs. This one is from chapter 1, 2.3. Let $X$ be a complete connected reduced ...
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253 views

Gorenstein ring VS. Gorenstein singularity

A normal variety is said to have Gorenstein singularity iff its canonical divisor is a Cartier divisor (one can always define the canonical divisor on a normal variety and it can be proved to be a ...
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94 views

Why $H^{1,1}(X,\mathbb{C}) = Pic(X) \otimes \mathbb{C}$ for Calabi-Yau 3-folds?

Let $X$ be a Calabi-Yau 3-fold, that is $\omega_X = 0$ and $h^{1,0}=h^{2,0}=0$. Let $\operatorname{Pic}(X)$ be the group of line bundles on $X$. Then why the following isomorphism is true ...
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49 views

Morphisms of affine line into a curve of higher genus

Let $X$ be a smooth curve of genus $\geq 1$ over a field $k$. How does one show that any $k$-morphism $f: \mathbb A^1_k \to X$ has to be constant? In general, is it true that any morphism $Y \times ...
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164 views

Every finite map is surjective

I'm trying to understand the proof of the theorem which states that every finite map is surjective in Shafarevich's book: I didn't understand why the part underlined in red is true. I need a ...
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1answer
811 views

Solutions to “Vakil - Foundations of Algebraic Geometry” exercises [closed]

I think the notes of Professor Ravi Vakil are a great source to learn algebraic geometry. The exposition is very clear, and much effort was put to give an intuitive picture together with a flawless ...
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166 views

Is Serre - Local algebra a good reference for an algebraic geometer?

Of course the question "what is a good commutative algebra book for an algebraic geometer" has been asked before, see A good commutative algebra book and Reference request: introduction to commutative ...
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153 views

Image of a diagonal morphism.

I was trying to study the definition of Sheaves of Differentials of Hartshorne p.175. It says The diagonal morphism $\Delta:X \rightarrow X \times _Y X $ gives an isomorphism of X onto its image, ...
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109 views

Tensor product, Artin-Rees lemma and Krull intersection theorem

I asked another question about tensor product, but can't conclude from the answer, so here is another more concrete question. Let $(A,m)$ be a local ring then by Artin-Rees Lemma $m^k \bigcap I ...
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74 views

Algebraical Fujita cancellation

If there exists some birational equivalence between ruled surfaces $C\times\mathbb{P}^1$ and $C'\times\mathbb{P}^1$ over a sufficiently nice field, then one can cancel the projective line and conclude ...
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115 views

The importance of being Cohen-Macaulay

I am starting to study Cohen-Macaulay rings, mainly from Bruns-Herzog book. In that book there are many examples and sentences of the type "If something satisfies this properties, then it is ...