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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69 views

About morphism from elliptic curve to projective space and pullback divisor

I'm sorry if the following question is trivial or even if it doesn't make any sense (I'm still learning about divisors). Suppose that $E/k$ is an elliptic curve, then the divisor $nO$ on $E$ is ...
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1answer
39 views

Singular homogeneous polynomial [closed]

Let $p$ be a homogeneous polynomial in $\mathbb{K}[x_1, \dots, x_n]$, where $\mathbb{K}$ is an algebraically closed field. If $\frac{\partial p}{\partial x_1}, \dots, \frac{\partial p}{\partial x_1}$ ...
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46 views

Is this polynomial irreducible over $\overline{\mathbb{Q}}$?

Let $d$ be a natural number greater than or equal to $3$. I have the following polynomial $$ F(x_1, ..., x_3) = x_1^d + x_2^d - x_3^d - ( x_1 +x_2 - x_3 )^d. $$ I am trying to figure out if this ...
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71 views

Proof of the formula that computes the genus of smooth projective plane curve

I was searching for a proof of the formula that computes the genus of a smooth projective plane curve of degree $d$: $$g = \frac{(d-1)(d-2)}{2}$$ which do not make use neither of triangulation or ...
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1answer
68 views

For each $g$ there is $[C]\in\mathcal{M}_g$ which embeds on a K3 surface

For each genus $g\geq 0$ there is a (smooth irreducible) curve $C$ of genus $g$ which embeds on some K3 surface. How does this follow from the surjectivity of the period map for K3 surfaces? Is ...
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1answer
54 views

Induced map on cohomology

Let $M= \mathbb P^1_{\mathbb C} \times \mathbb P^1_{\mathbb C} \times \mathbb P^1_{\mathbb C}= S^2 \times S^2 \times S^2$, and let $D$ be the subvariety of $M$ defined by the equation $$ t_0t_1t_2 = ...
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1answer
45 views

Existence of categorical quotient $X/\mathbb{G}_{m,A}$.

Let $A$ be an $\bar{\mathbb{F}}_p$-Algebra of finite type (one might assume $A$ to be reduced). Let $X \subset \mathbb{A}_A^d\backslash \{0\}$ be a closed $A$-subscheme together with a group action of ...
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43 views

Explicit equations for morphism determined by a linear system, Part II

This is a follow-up to this question: Explicit equation for extension of a rational map? . I realized my wording was a bit unclear and that I could have been more direct in what I was asking for. ...
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1answer
65 views

Reference request: Fibre functor for elliptic curves is pro-representable

I am writing a project on étale fundamental groups of elliptic curves and I want to include a proof of a key theorem: the fibre functor on the category of finite étale covers of an elliptic curve is ...
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1answer
31 views

Sum of orders is zero, function field

Let $f \in \mathbb{C}(X)^\times$. Does it necessarily follow that$$\sum_{v \in \mathbb{CP}^1} \text{ord}_v(f) = 0?$$Here, $\text{ord}_v$ denotes the order of zero of $f$ at $v$. Update. $C(X)^\times$ ...
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32 views

Total transform of an effective Cartier divisor in blow-up, Vakil's Ex 22.4.P

Suppose $X$ is a scheme (assume it is Noetherian if necessary) and $p$ is a closed point whose sheaf of ideals is of finite type, then the blow up of $X$ along $p$, $\text{Bl}_p X$, exists and could ...
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1answer
36 views

About Elliptic Fibration. A point of generic fibre corresponds to a section.

I am reading Schutt and Shiva's Elliptic Surfaces. Suppose $f:S\longrightarrow C$ is an elliptic fibration, where S is a smooth projective surface and C is a smooth projective curve. Then the generic ...
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31 views

Line bundles on affine schemes

If $A$ is a UFD, all the line bundles on $\text{Spec}~A$ is trivial. Is there any generalizations of this about when the Picard group of an affine scheme is trivial? Or are there any theorems about ...
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66 views

For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $\mathcal{O}_X$-module $F$ has the structure of a $G$-representation

Reading Huybrechts & Lehn's "The geometry of moduli spaces of sheaves" I am stuck with a particular statement that they make in the chapter on GIT without explanation. Namely, they say that for an ...
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1answer
38 views

Smoothness of the intersection of smooth varieties

Let $(L_i)_{i \in I}$ be a finite family of (distinct) smooth irreducible embedded algebraic varieties of codimension one in some complex projective space of dimension $n$. Consider a finite ...
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39 views

Closure of algebraic groups

Let $\phi: G\rightarrow V$ an embedding, with $G$ a complex algebraic group and $V$ a vector space (actually a $G$-representation). Is it true that the closure (in the Zariski topology) of $\phi(G)$ ...
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32 views

Checking a global property over the closed point of a local ring

Let $k$ be a field and let $T=\textrm{Spec }R$ be the spectrum of a local $k$-algebra $(R,\mathfrak m)$. Let $X\to T$ be a proper flat map from a $k$-scheme $X$, and suppose given a morphism of ...
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49 views

Example of a complete non-projective variety

I am attempting to solve hartshorne's exercises. I am stuck at the exercise 7.13(b) of chapter 2 which is the following Let $k$ be an algebraically closed field of char $\neq$ 2.Let $C \subset ...
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1answer
43 views

If $\phi: X \rightarrow Y$ is a morphism of varieties, then $\phi(X)$ contains a nonempty open subset of $\overline{\phi(X)}$.

This question has been asked before, but I am stuck on a different part. If $\phi: X \rightarrow Y$ is a morphism of varieties over an algebraically closed field $k$, I'm trying to understand why ...
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36 views

Exercise 2.3.17 from Qing Liu

This is exercise 3.17 from Qing Liu's Alg.Geo. book. Let $f:X\rightarrow Y$ be quasi-compact morphism of scheme. Let $\mathscr{J}=Ker f^{\sharp}$ and let $Z=V(\mathscr{J})$. Show that $Z$ is a ...
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1answer
27 views

A confusion with the Definition of 'Skyscraper Sheaf' from 'Stack Project'

According to Hartshorne, (Chapter, Ex 1.17): Let $X$ be a topological space, let $P$ be a point, and let $A$ be an abelian group. Define a sheaf $i_p(A)$ as follows: $i_P(A)(U)=A$ if $P\in U$ and ...
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1answer
26 views

Flatness in terms of the freeness of the pushforward of the structure sheaf…

Ravi gives a version of the claim below in exercise 24.4G in "Foundations...", but he adds the condition that $Y$ is reduced in order to conclude something additional (the equivalence of the condition ...
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1answer
46 views

The equivalence of two definitions of closed subscheme, Vakil's Ex 8.1.K

Generally in literature, the definition of a closed embedding in the category of scheme is a morphism $\pi:X \rightarrow Y$ between two schemes such that $\pi$ induces a homeomorphism of the ...
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1answer
47 views

When $k$ is algebraically closed a regular point is smooth

I would like to show that in the case of a $k$-scheme of locally finite type with $k$ algebraically closed, a regular point is smooth. For smooth implies regular, the proof in Görtz and Wedhorn is ...
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2answers
25 views

Show that some set is an algebraic set

Show that $\{(t,t^2,t^3):t\in k\}$ for a field $k$ is an algebraic set. Just by looking at the points, i see that they are zeros of $F_1(x,y,z)=xy-z$, $F_2(x,y,z)=xz-y^2$ and $F_3(x,y,z)=yz-x^5$. I ...
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1answer
37 views

Why “to hold in a nonempty Zariski open set” implies “to hold for a random sequence”

Let $k$ be an algebraically closed field and let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables. Let $d_1,...,d_m$ be non-negative integers. For each $d$, consider the linear subspace ...
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1answer
39 views

Hartshorne 4.1.6 Gonality of a curve

I have a question about the following exercise from Hartshorne's book 'Algebraic geometry': Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f:X\rightarrow \mathbb P^1$ with ...
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1answer
36 views

Nakayama and locally free sheaves

Let $S$ be a noetherian scheme and $F,E$ be two coherent sheaves on $S$ with $E$ locally free. Suppose we have a morphisme $f : F \to E$ such that $f_s : F \otimes \kappa(s) \to E \otimes \kappa(s)$ ...
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1answer
71 views

Lines on a Quintic Threefold

We work over an algebraically closed field $k$. I've been given the exercise of showing (using only the technology introduced in the first chapter of Shafarevich's Basic Algebraic Geometry) that ...
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66 views

Representing intersections of Cartier divisors

For a nonsingular variety $X$ over the complex numbers, take two Cartier divisors $D,E$ determined by the data $(U_i, f_i)$ and $(U_i, g_i)$. The intersection $D \cdot E$ should be represented by a ...
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Can I choose $k+1$ hypersurfaces to avoid a fiber of dimension $k$ in projective space?

Let $X$ be a closed subscheme of dimension $k$ in $\mathbb{P}^n_A$, where $A$ is a Noetherian ring. In Exercise 11.3.C of Ravi Vakil's notes, it is shown that one may choose $k+1$ hypersurfaces such ...
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18 views

Rings of Regular functions, and regular maps between Quasi Affine to Quasi Proj. Varieties.

I have studied classical algebraic geometry a while ago. I want to sum up in short as possible everything regarding their rings of regular functions. If my understanding not correct, please correct ...
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1answer
25 views

Details about the definition of: “deformation of a family”

Let $f:X\to Y$ be a flat, surjective morphism of $k$-schemes with connected fibres i.e. $f$ is a family. Definition: Let $T$ be a $k$-scheme. A deformation of $f$ (over $T$) is a family $g:\mathfrak ...
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1answer
33 views

Sections of projective bundles

Suppose we have a vector bundle $\pi:\text{Spec}~A[x_1,\cdots,x_n] \rightarrow \text{Spec}~A$, then the sections are morphisms $s:\text{Spec}~A \rightarrow \text{Spec}~A[x_1,\cdots,x_n]$ such that ...
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2answers
89 views

Geometric reasons finite fields have prime power orders?

All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
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37 views

An irreducible variety is not composed of finitely many subvarieties

There is a lemma in commutative algebra: Let $\mathfrak{a}_1, \dotsc, \mathfrak{a}_n$ be ideals such that $\mathfrak{a}_n \cap \dotsb \cap \mathfrak{a}_n$ is contained in a prime ideal ...
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1answer
39 views

Product of a variety with a rational variety

Let $X$ and $Y$ be two algebraic varieties such that $X$ is rational and $X\times Y$ is unirational, do this implies that $Y$ is unirational as well?
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1answer
37 views

The stalk of a specialization is a localization of the generization.

So I am trying to prove that in for a scheme locally of finite type over an algebraically closed field, smoothness and regularity are the same thing. I am working in Görtz and Wedhorn. In lemma 6.26, ...
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2answers
70 views

Commutation of pushforward and pullback along cartesian squares

I am sure this is well-known, but I cannot find a reference. Consider a cartesian square $$\require{AMScd} \begin{CD} P @>{v}>> X\\ @V{g}VV @VV{f}V \\ Z @>{u}>> Y \end{CD}$$ where ...
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20 views

Toric variety associated to a rotated fan

Let $\Delta$ be a fan in the lattice $\mathbb Z^2$ consisting of SCRAP cones and let $X(\Delta)$ be the associated toric surface. Suppose we rotate fan in the counter-clockwise direction by a rational ...
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1answer
29 views

describing proj. seurface.

I have the surface $W=Z(x_0x_1-x_2x_3)$, in $\mathbb{P^3}$ and I want to describe it as a union of an affine piece and some other piece laying in $\mathbb{P^2}$. My solution is to look at: ...
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1answer
37 views

Dimension and intersection of algebraic varieties

Let $L,M$ be projective varieties over $\mathbb C$ of same dimension. 1) Assuming $L$ and $M$ are embedded hypersurfaces in $\mathbb{P}^n$, is it true that $L$ and $M$ must have non-empty ...
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2answers
45 views

Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or ...
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1answer
15 views

Showing a morphism is birational

$~$ Hey y'all, I was wondering if you could help me with the following: let $f: \mathbb{A}^1\rightarrow V(x^2-y^3,y^2-z^3)$ be the map defined by $f(a)=(a^9,a^6,a^4)$. How can I show that $f$ is ...
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1answer
52 views

Rational Maps Between Curves

Let $F:C'\rightarrow C$ be a rational map. Then either $F$ is dominating, or $F$ is constant. Furthermore, if $F$ is dominating, then $k(C')$ is a finite algebraic extension of $\tilde{F}(k(C))$. Why ...
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1answer
46 views

Hartshorne Lemma V.1.3 meaning of exact sequence

I've been trying to make sense of the exact sequence in Lemma 1.3 chapter 5. The Lemma is the following: Let $C$ be a smooth irreducible curve on a smooth projective surface X, and let $D$ be any ...
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1answer
90 views

What do the closures of cyclic groups in $\textrm{GL}_n$ look like?

Let $k$ be algebraically closed, $G = \textrm{GL}_n$ in the Zariski topology, and let $g \in G$. Let $H$ be the subgroup generated by $g$. Assume that $g$ does not have finite order. Question: ...
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1answer
24 views

Completion of a local ring, Vakil 29.3A

If $p$ is a point of $X$, which is a $\bar{k}$ variety of dimension $1$, $p$ is a node if the completion of $\mathcal{O}_{X,p}$ at $m_{X,p}$ is isomorphic to $\bar{k}[[x,y]]/(xy)$. If now ...
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0answers
23 views

Verlinde Formula in geometric quantization?

I think I have a fair grasp on the $\rm{SU}(2)$ Verlinde Formula from the algebraic geometry perspective. I'm hoping to understand better how exactly this relates to the geometric quantization of a ...
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0answers
17 views

Relation between moduli space of flat connections and moduli space of bundles on curves

I would like to let $X$ be a genus $g$ curve, with $M_{g}(r,d)$ the moduli space of bundles on the curve such that $(r,d)=1$. Alternatively, we can pick a group $G$, consider principal $G$-bundles ...