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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Differential of boundary morphisms in the moduli space of pointed stable curves.

Recall that first order deformations of a smooth pointed curve $(C,p_1,\ldots,p_n)$ are parametrized by $H^1(C,\cal{T}_C(-p_1-\ldots-p_n))$ and in the stable case is ...
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1answer
411 views

Does there exist a regular map $\mathbb{A}^1\to\mathbb{P}^1$ which is surjective?

Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields. I'm ...
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34 views

Intersection of affine open subschemes

If X is separated scheme, then for any affine opens U, V, the intersection is affine. If X is quasi-separated, then the intersection can be covered by finitely many affine opens. Is there some ...
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1answer
39 views

Explicit description of the inverse image sheaf of an ideal sheaf.

$\DeclareMathOperator{\Spec}{Spec}$ Let $f: \Spec A \to \Spec B$ be a morphism of affine schemes and $f^\#: B \to A$ be the corresponding ring homomorphism. Let $\mathcal{I} \subseteq ...
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1answer
30 views

$div(z)=0\Leftrightarrow z\in k$

I'm reading algebraic curves book from Fulton and I didn't understand this corollary on page 98: Why $\deg(div(z-\lambda_0))\gt 0$? and why is this a contradiction? Thanks a lot
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1answer
83 views

Hilbert Series of $\mathbb{C}^2$?

Consider the following ideal in the polynomial ring $\mathbb{C} [x,y,z]$: \begin{equation} I = \langle z^2, yz \rangle \end{equation} One can compute the Hilbert series of the affine varieties ...
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32 views

Number of parameter of a quadric

Suppose for example that $S$ is an algebraic complex surface contained in $\mathbb{P}^6$. $S$ is the complete intersection of four quadrics in the six dimensional projective space. If i take a quadric ...
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1answer
23 views

What is precise definition of $monomial~curve$ in affine e-space?

What is precise definition of $monomial~curve$ in affine e-space ?
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1answer
22 views

Equivalence of points on a smooth cubic in $\mathbb{P}^2$

Let $C \subset \mathbb{P}^2 (\mathbb{C})$ be a smooth cubic. Show that $(p)$~$(q)$ if and only if $p=q$. $p$ and $q$ are points, and two divisors $D$ and $D'$ are ~ if $D-D' = (f)$ for some ...
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41 views

Fano surfaces all of whose rational points lie on some geometric line

Are there any ? Namely let $X$ be a smooth del Pezzo surface defined over $\mathbb{Q}$ that has rational points and such that the degree of the del Pezzo is small, say $d=3$ or $4$. Is it possible ...
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38 views

Visualizing Curve in Projective Space

I'm wondering if anyone can help me understand/visualize what the curves xy=1 and $y=x^3$ look like in projective space. I'm familiar with the construction of the space, and how to homogenize the ...
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1answer
40 views

Automorphisms of non-hyperelliptic curve of genus 3 in $\mathbb{P}^{2}$

I have a question from R. Vakil's exercise 19.7.C which goes as follows: Suppose $C'\subset\mathbb{P}^{2}$ is a smooth plane quartic curve. Show that there is bijection between automorphisms of $C'$ ...
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113 views

Moduli space of algebraic surfaces Vs moduli space of curves

Define the surface $S$ as the complete intersection of four quadrics $Q_i$ with $i=1,2,3,4$ in $\mathbb{P}^6$ (complex six dimensional projective space) i.e. $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4$$. Put ...
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1answer
56 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
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57 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
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35 views

Exercise on Fulton's “Algebraic Curves”

Exercise 7.12 from Fulton's Algebraic Curves Find a quadratic transformation of $\; F = Y^2 Z^2 − X^4 −Y^4$ with only ordinary multiple points. By checking the partial derivatives, I found that ...
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39 views

Better understanding regular functions on a Projective variety

Hi guys I was just looking an example from class that was left as obvious, but it is not so obvious to me. $W= V(x_1x_4-x_2x_3)= $ where $I(W)= \langle x_1x_4-x_2x_3 \rangle$ so we just picked an ...
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1answer
102 views

group law of complex torus is divisible?

I need help with this exercise: Show that the group law of a complex torus (the definition I have is that of Rick Miranda's book Algebraic curves and Riemann surfaces, the one that he constructs from ...
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1answer
43 views

Why the kernel of the restriction map $\text{Pic}^0(X)\to \text{Pic}(U)$ is finitely generated?

Let $X$ be a smooth and geometrically connected projective curve over a field $k$ and $U$ be a non-empty open subset of $X$. Let $\text{Pic}^0(X)$ be the degree $0$ Picard group of $X$ and ...
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1answer
461 views

Homeomorphism between complex torus and S1 x S1

We have the lattice L={$m_1w_1 + m_2w_2 | m1,m2 ∈ \mathbb Z, w1,w2 ∈ \mathbb C $}. We want to construct an homeomorphism between $\mathbb C/L$ and $\mathbb S^1 \times \mathbb S^1$. I've read that the ...
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43 views

What is meant by the discriminant locus of a fibration?

If we have a fibration $f: X \to B$ (allowing singular fibers) of a differentiable manifold $X$, precisely what is meant by "the discriminant locus of $f$ " $\Delta \subset B$ and how do we define it? ...
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37 views

Computing singular locus

I should compute the singular locus of $V(x^n+y^k)$ where $k,n\ge 1$ are natural numbers and $f=x^n+y^k\in K[x,y]$. Here $K$ is a field of characteristic $p$ (which means, as far as I understood it, ...
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45 views

Birational morphism $\mathbb{P}^n \to \mathbb{P}^n$

Let $f: \mathbb{P}_{\mathbb{C}}^n \to \mathbb{P}^n_{\mathbb{C}}$ be a birational morphism. Question: Is $f$ necessarily an isomorphism? I know that if $f$ is in addition to the assumptions above ...
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2answers
47 views

Smoothness of $A \subseteq C$ implies smoothness of $B \subseteq C$? where $A\subseteq B \subseteq C$

Let $A \subseteq B \subseteq C$ be commutative rings (noetherian integral domains, if this helps). Assume $C$ is a smooth $A$-algebra. Is it true that $C$ is a smooth $B$-algebra?
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1answer
41 views

Formula for top self intersection of exceptional divisor

Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $\pi: Y \to X$ be the blow-up of a smooth point $x \in X$. Is there a nice formula for the intersection number $E^n$?
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11 views

Is there a pluricanonical divisor on a relatively minimal complex elliptic surface that can be written as sum of fibres?

A complex algebraic surface $S$ is said to be elliptic if there are a smooth curve $B$ and a surjective morphism $p \colon S \to B$ whose generic fibre is an elliptic curve (i.e. a smooth curve of ...
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0answers
21 views

Centers of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
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1answer
51 views

Ideal sheaf is aquasi-coherent sheaf of ideals.

Let $X$ be a scheme, For any closed subscheme $Y$ of $X$, the corresponding ideal $I_Y$ given by the kernel of the morphism $i^{\#}:\mathcal{O}_x\rightarrow i_{*}\mathcal{O}_y$ is quasi-coherent sheaf ...
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Regarding the connected component of $|1/J| < 1$ containing $\infty$

How does one explicitly describe the connected component of $|1/J| < 1$ containing $\infty$? Here, $J = J(\tau) = j(\tau)/12^3$ is the normalized $j$-invariant so that $J(i) = 1$, and $\tau$ is in ...
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1answer
38 views

Varieties and subsets

I'm studying for my last exam and I got stuck in this exercise, from Fulton of Algebraic Geometry. Let $V$ be an affine variety, $f\in \Gamma(V)$ a) Prove that $V(f) = \{P\in V | f(P)=0\}$ is a ...
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1answer
75 views

Simple question about extending morphisms to $\mathbb{P}^1$

A trivial question, but my lack of working experience in algebraic geometry is a hurdle. Show that every morphism from $\mathbb{A}^1-\{0\}$ to $\mathbb{P}^1$ extends to a morphism from ...
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38 views

Intersection of nef divisors

Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $D_1,\cdots,D_n$ be nef Cartier divisors on $X$. Is it true that $(D_1 \cdots D_n)^n \geq D_1^n \cdots D_n^n$? I know that ...
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1answer
39 views

If $C \subseteq \mathbb{P}^2$ is a plane curve, then $genus(C)=\frac{1}{2}(d-1)(d-2)$. Compare with example in the notes

In my Algebraic Geometry notes (see http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf) there is the following exercise: If $C \subseteq \mathbb{P}^2$ is a plane curve of degree ...
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40 views

pushforward/pullback map of sheaves

Let $f:X \rightarrow Y$ be a morphism of schemes. Let $F$ be a sheaf on $Y.$ When is it true that the natural map $F \rightarrow f_* f^* F$ is an isomorphism?
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1answer
41 views

Deriving the Quadratic Polynomials Defining the Twisted Cubic

I've recently been reading about rational normal curves and how they may be represented and have come to the following question: (for simplicity's sake, the problem is stated in terms of the RNC in ...
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1answer
134 views

Irreducibility of the $k$-secant variety of the Veronese variety.

I have read that the $k$-secant variety $\sigma_{k}(V_{d}^{n})$ of the Veronese variety $V_{d}^{n}$ is irreducible, but I do not know how to prove it. If $k=2$, it is clear, because the variety of ...
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0answers
23 views

regular function defined on $V(F)$ without using closed map of morphisms on projective varieties.

Let $k$ be an algebraically closed field and let $F\in k[x,y]$ be an irreducible polynomial. I want to prove that if $z$ is a function regular on all the points of $X=V(F)$ then $z$ is constant. I ...
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1answer
20 views

homotopy module of a simplicial module

I'm reading a paper about the cotangent complex and I'm having trouble with one of the definitions (3.4 of http://homepages.math.uic.edu/~bshipley/iyengar.pdf ). Let $V$ be a simplicial $R$-module. ...
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39 views

Kähler differentials, define valuation?

See my previous question for a definition of the $K$-module of Kähler differential $\Omega_{K/k}$. This question is sort of a follow up on it. Suppose $k$ is a field of characteristic $0$, $R$ is a ...
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2answers
22 views

Finding Coefficient given 2 Equations of Lines and an Angle

A line has equation $$3x - ky = 0$$ Find the value of k if this line makes an angle of 45 degrees with the line $$2x + 5y - 17 = 0$$ The answer among the choices is supposed to be $7$. But I keep ...
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75 views

Exercise $1.8$ of chapter one in Hartshorne.

In exercise 1.8 of chap I in Hartshorne algebraic geometry, Let $Y$ be an affine variety of dimension $r$ in $\mathbf A^n$. Let $H$ be a hypersurface in $\mathbf A^n$, and assume that $Y ...
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Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
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1answer
39 views

Step in the construction of the global spec of a sheaf of algebras

I'm working my way through the construction of the global spec of a sheaf of algebras. Here is the setup. Let $ Y $ be a scheme. Let $ \mathscr{A } $ be a quasi coherent sheaf of $ ...
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56 views

What is the geometric interpretation of a P-primary component of an affine k algebra?

Let $R= K[x_1, x_2,...,x_n]$ for some algebraically closed field K. If $I \subset R$ is an ideal, and P is a prime minimal over I, I know that $Z(P)$ is a maximal irreducible subset of $Z(I)$. But ...
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0answers
27 views

What is the definition of Osculating plane in algebraic geometry?

I'm studying Fulton's algebraic curves book and in order to understand this paper in Algebraic Curves I need the definition of the d-dimensional osculation plane. Can I understand properly this ...
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1answer
48 views

Equivalence of line bundles and $\mathbb{G}_m$-torsors

This appears to be a duplicate of (half of) this question, but it received no attention so I'll try again. Given a line bundle $L\to X$ on a scheme $X$ over a field $k$, I am to show that ...
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0answers
19 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
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0answers
25 views

Properties preserved by fppf morphisms

Which properties P do fppf morphisms preserve? In other words if $f: X \to Y$ is fppf and $Y$ has P, for which P does $X$ also have P? I'm particularly interested in the cases when P=smooth or ...
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1answer
31 views

Isomorphic projective subvarieties, non-isomorphic rings

If $S \subset \mathbb{P}^n$ is a closed set (in the Zariski sense) then $\mathcal{I}(S) \subset k[x_0,\ldots,x_n]$ denotes the homogeneous ideal of polynomials which vanish at $S$. I want to find an ...
4
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1answer
45 views

Blow-up of pair of intersecting lines

I have the reducible variety $X=\mathbb{V}(x_1x_2)\subset\mathbb{A}^2$, which is a pair of lines intersecting transversely, and I would like to compute the blow-up at the origin. The Rees ring of the ...