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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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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53 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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107 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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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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146 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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183 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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966 views

Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
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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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239 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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92 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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163 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
785 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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161 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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148 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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108 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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112 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 ...
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128 views

Generic Points to the Italians

When I first learned algebraic geometry, I naturally wiki-ed the subject and there was a line there that said the old school Italians used the notion "generic points without any precise definition." ...
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130 views

Can gluing of morphisms of locally ringed spaces be expressed by an exact sequence?

Suppose $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ are locally ringed spaces. Then morphisms glue, that is, if $\{U_i\}_i$ is an open cover of $X$, then "to give a morphism $X\to Y$ is the same as ...
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131 views

Reference about Moduli Spaces of Riemann surfaces

I'm looking for some (introductory, and in any case not too technical) reference (book, lecture notes, papers) regarding moduli spaces $\mathcal{M}_g$ and $\mathcal{M}_{g,n}$ of (punctured) Riemann ...
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42 views

Lifting extensions of sheaves

Consider a curve $X_0$ over an algebraically closed field $k$. Let $A$ be a local Artinian ring over $k$ and $X$ a scheme over $A$ with $X \otimes_A k = X_0$. For a sheaf $F$ on $X$ we denote by $F_0$ ...
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59 views

Vanishing of sections and special divisors

Let $L$ be a line bundle on a smooth complex projective curve $X$. Suppose we have vector subspaces $$U\subset V\subset H^0(X,L),\,\,\,\textrm{and}\,\,\,\dim\, U\leq k,\,\,\dim\,V=k+1.$$ I wonder if ...
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117 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
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80 views

A characterization of Henselian rings

It is well known that if $(A, \mathfrak m)$ is a Henselian local ring with residue field $\kappa$, then base-change from $A$ to $\kappa$ determines an equivalence of categories $$F: \{\text{Finite ...
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130 views

(Graded) deformations of algebras

I'm reading the article of Braverman and Gaitsgory, Poincaré-Birkhoff-Witt theorem for quadratic algebras of Koszul type, but I'm stuck in a point near the beginning. Let $A$ be a (positively) graded ...
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95 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
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80 views

Computing an integral basis of an algebraic function field, $y^4-2zy^2+z^2-z^4-z^3=0$.

I am trying to compute an integral basis for the algebraic extension $K(z,y)$ of $K(z)$ by $y$, with $f(z,y)=0$, $$ f(z,y) = y^4-2zy^2+z^2-z^4-z^3 = 0. $$ $K$ here is either $\mathbb{Q}$ or ...
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109 views

What does “generic” mean in this context, and is it related to generic points in algebraic geometry?

In "The characteristic polynomial and determinant are not ad hoc constructions" by Skip Garibaldi, available at http://arxiv.org/abs/math/0203276 the characteristic polynomial is defined as the ...
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78 views

The importance of generating series

What follows is a very nebulous question. I just seek for some help in understanding a "technique" which has proven itself very powerful. I noticed that very often generating series appear in ...
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82 views

About the Stein factorisation of a morphism defined by a complete linear system on a surface, whose image is a curve

Let $X$ be an algebraic complex projective surface, and let $D$ be an effective divisor on $X$ with empty base locus. Assume that the morphism $\varphi = \varphi_D : X \to \mathbb P^{h^0(D)-1}$ has ...
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325 views

Cardinality of the Fiber of a Finite Morphism Vs. Degree (via Vakil)

I would like to show the following (note: it is not an assigned problem, so it may be false) (EDIT: Indeed it is false, see end of post): Suppose $f:X \rightarrow Y$ is a finite, surjective morphism ...
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149 views

Fundamental group of a space under a group action

The short version: Why does a Borcea-Voisin threefold has trivial fundamental group? Well, what is a Borcea-Voisin threefold? These are named after Borcea and Voisin, who introduced a method of ...
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127 views

Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
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1answer
155 views

On a Geometric Interpretation of the Local Criterion for Flatness in Eisenbud's

The local criterion for flatness goes this way: Let $\phi : (A,m)\rightarrow (B,m')$ be a local morphism of local Noetherian rings, and $M$ a finitely generated $B$-module. If $x\in m$ is a non ...
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1answer
176 views

Terminology for blow-ups in algebraic geometry

Definition (from Eisenbud-Harris' Geometry of Schemes): Let $X$ be any scheme, $Y \subset X$ a subscheme. We say that $Y$ is a Cartier subscheme in $X$ if it is locally the zero locus of a single ...
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84 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that ...
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44 views

Properties of divisors when moving from char 0 to char p.

Consider a smooth projective variety $X$ over $\mathbb{C}$ such that $X$ has models over $\mathbb{Z}[1/N]$ and $X_p=X_{\mathbb{Z}[1/N]}\times \text{Spec}(\mathbb{F}_p)$ is also a smooth projective ...
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1answer
165 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
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141 views

Modular Functions with Rational Fourier Expansions

I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
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40 views

$H^{1}(O_{F})$ of a surface in a toric variety

I have a surface inside a toric variety $X$ and I would like to compute the first cohomology of its structure sheaf via the Cech complex, since I already know which cones of $X$ it hits (five ...
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53 views

Existence of Solutions of Two Cubic Equations in a Particular Region

If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
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118 views

Computing a contraction of an exceptional divisor.

For a few days, I have been working on the following problem, from Qing Liu's book: Let $\mathcal{O_K}$ be a discrete valuation ring with uniformizing parameter t and residue characteristic $\neq ...
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99 views

serre duality and logarithmic differentials

Let $D$ be a normal crossings divisor on some smooth projective variety $X$ (say over the complex numbers) and let $\Omega^p_X(\log D)$ be the sheaf of logarithmic $p$-forms. It is $$ \Lambda^p ...
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1answer
63 views

Dimension of range of an function

Let $f$ be a rational function from affine variety $X$ to affine variety $Y$. Is it always true that $\dim X \geq \dim f(X)$? If it is can someone provide me with a proof of it? To me, this is ...
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391 views

Gluing schemes Hartshorne example

In example 2.3.5 Hartshorne says Let $X_1$ and $X_2$ be schemes. Let $U_1 \subseteq X_1$ and $U_2 \subseteq X_2$ be open subsets, and let $\varphi: ( U_1, \mathcal{O}_{X_1 \mid U_1}) \to ( U_2, ...
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1answer
108 views

Counter example of upper semicontinuity of fiber dimension in classical algebraic geometry

We know that if $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each point of $X$ the dimension of the ...
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79 views

Does “low dimension” of a constructible set of $\mathbb C^n$ imply “zero measure” of the set in $\mathbb C^n$ and $\mathbb R^n$?.

I know that if some specific statement $S$ holds for $x\in\mathbb C^n$, then $x$ belongs to a constructible set $X\subset C^n$ (in Zariski topology) of dimension strictly lees than $n$. ${\bf ...