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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62 views

Calculating object position in 3D space

I'm looking for an algorithm to calculate the position of point P in space using a triangular(/rectangular) plane on the 'ground'. The position between the points ABC of the triangle on the ground are ...
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1answer
30 views

Determining irreducible components of $Z(y^4 - x^6, y^3 - x y^2 - y x^3 + x^4) \subset \mathbb A^2$

I'm trying to determine the irreducible components of the zero set $Z(I)$ for the ideal $I = (y^4 - x^6,\, y^3 - x y^2 - y x^3 + x^4)$, in the affine space $\mathbb A^2$ over an algebraically closed ...
4
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2answers
57 views

Geometric Intuition Behind Blowing Up a Cusp on a Plane Curve?

I'm reading Hartshorne AG V.3 on monoidal transformations and embedded resolutions. I understand one sort of intuition behind blowing up a point on a surface (or more generally a subvariety of a ...
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0answers
40 views

short exact sequences of complexes and triangles in the homotopy category

Suppose I start with an abelian category $\mathcal{A}$, form its category of complexes $C(\mathcal{A})$ and consider a short exact sequence in this category: $$0 \to A^{\bullet} \to B^{\bullet} \to ...
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1answer
40 views

Equivalence definitions of hyperelliptic curves

I'm reading Fulton's algebraic curves book and on page 111, he defines hyperelliptic curves. For him an hyperelliptic curve $C$ is a curve which has a hyperelliptic weierstrass point $P$, i.e., $2$ is ...
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1answer
101 views

Calculating global sections of sheaves

Consider the usual projective space $\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}$, and the Weil divisor $D = \{0\} \subset \mathbb{C} \subset \mathbb{P}^{1}$. Writing projective space as the union of ...
3
votes
1answer
71 views

Translation from schemes to varieties

At the moment, I know very little algebraic geometry (sadly!) so I apologise for the silliness/stupidity of these questions. Set up: Let $k$ be a field (not necessarily algebraically closed). Take ...
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1answer
40 views

How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one ...
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2answers
104 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
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2answers
96 views

Counting the number of solutions of equation $x^2 + y^2 = 1$ over $\Bbb Z/p$

List proofs of the fact that the number of solutions to $x^2 + y^2 = 1$ over $\Bbb Z/p$, where $p$ is a prime $\neq 2$, is $p-(-1)^{\frac{p-1}2}$. I thought of two. I write one below.
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55 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
4
votes
1answer
67 views

Maximal ideals of polynomial ring

We know that if $k$ is algebraically closed, then each maximal ideals of $k[x_1, x_2, \ldots , x_n]$ are of the form $(x_1 - a_1, x_2 - a_2, \ldots, x_n - a_n),$ where $a_1, a_2, \ldots , a_n \in k$ ...
2
votes
1answer
33 views

quotients by quasi-coherent ideal sheaves

I saw this lemma stated in some lecture notes: If $\mathcal{I}$ is a quasi-coherent sheaf of ideals on a scheme $X$ and if $U$ is any affine open subset of $X$, then ...
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1answer
43 views

Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property

Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on ...
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0answers
56 views

Help in a proof in Fulton's algebraic curves book

I'm reading Fulton's algebraic curves book and I didn't understand this proof of proposition 7 (page 106) very well: So I have the following doubts: I didn't understand why $\text{ord}_P(f')\ge ...
2
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1answer
36 views

Connected component identification?

Suppose I give a random 2 variable polynomial relation such as: $$x^3+y^3=10$$ $$x^2 + 7yx^4 + x^2-15=0$$ Etc... How do I determine how many individual pieces there are to the graph?
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1answer
49 views

function field of $zy^2 - x^3$ in the plane

I am interested in understanding the connection between abstract curves and smooth projective curves. So I looked at a simple example $zy^2 - x^3$ in $\mathbb{P}^2$. The function field can be computed ...
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1answer
35 views

If a curve is hyperelliptic, we have an equality in Clifford's Theorem

I'm studying Fulton's algebraic curves book and I have the following question: Clifford's theorem says that if $D$ is a divisor and $W$ is a canonical divisor with $l(D)\gt 0$ and $l(W-D)\gt 0$, then ...
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0answers
29 views

Complex elliptic surface with 24 $I_1$ fibers

Is a complex elliptic surface with 24 $I_1$ fibers always a K3 surface? Is ti possible to characterize a K3 surface in terms of the singular fibers of a given elliptic surface?
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votes
1answer
40 views

functor of points for grassmannian

I am reading section 16.7 of Vakil's Foundations of AG, which constructs the Grassmannian G(k,n) via its functor of points, namely B -> surjections O^n_B -> Q, where Q is locally free of rank k. He ...
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0answers
50 views

Visualizing projective closures - is it okay to just think of the affine case?

This question is quite general and has been discussed on MSE before, however my case is a little bit different and I'm wondering about the geometric interpretation of a specific example. I think that ...
1
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1answer
49 views

Distinguished points of a cone

Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ ...
3
votes
1answer
42 views

Equivalence relation on regular functions

In this problem, consider $K$ an algebraic closed field and $X\subset\mathbb{A}^n_k$ an irreducible variety. Given an open Zariski $U\subset X$, we say that a function $\phi:U\rightarrow K$ is regular ...
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90 views

properties of pullback diagrams

Suppose you have a commutative diagram: $\require{AMScd}$ $\begin{CD} A @>>> B\\ @VVV @VVV \\ C @>>> D \\ @VVV @VVV \\ E @>>> F \end{CD}$ Let $T$ be the top "square", $B$ ...
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44 views

Jacobian matrix rank and dimension of the image 3

Let $p_1,\dots,p_n,q_1,\dots,q_n$ be polynomials in $m<n$ variables and $f(z)=\left(\frac{p_1(z)}{q_1(z)},\dots,\frac{p_n(z)}{q_n(z)}\right)$. By construction, the variety $V$ is the Zariski ...
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0answers
51 views

Are any two $\ell$-adic Tate twists (non-canonically) isomorphic?

Recall that for a prime $\ell$, the $\ell$-adic Tate twist is defined by $$\mathbb Z_{\ell}(n) := \varprojlim_r \mu_{\ell^r}^{\otimes n}.$$ As abelian groups, we have a (non-canonical) isomorphism ...
2
votes
1answer
57 views

Blowing up families of singular curves

I am stuck with a simple example, but I guess the more general question would be whether blow ups commute with restrictions to subsets (points) of the blow-up locus. Over $\mathbb{C}$, suppose that ...
5
votes
1answer
150 views

Algebraic geometry papers for beginners

What are some papers/books suitable for a beginning graduate student interested in algebraic geometry? I have taken commutative algebra and a classical algebraic geometry class, but I have no other ...
2
votes
1answer
93 views

Flatness of $\Omega_{B/K}$ over $B$.

Let $K$ be a field of characteristic zero. Assume that $K \subset A \subseteq B$ are noetherian integral domains, with $A$ regular (= all its localizations at maximal ideals are regular local rings). ...
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26 views

subvariety of $(\mathbb P^1)^4$.

Let $S$ be the sub-algebra generated by the set $S=\{ x_1x_2y_3y_4,\ x_1x_3y_2y_4,\ x_1x_4y_2y_3,\ x_2x_3y_1y_4,\ x_2x_4y_1y_3,\ x_3x_4y_1y_2 \}$ of homogeneous polynomials. I need to compute ...
3
votes
1answer
58 views

why does infinitesimal lifting imply triviality of infinitesimal deformations?

I'm trying to learn some deformation theory, but I'm stuck on the proof of corollary 4.7 in https://math.berkeley.edu/~robin/math274root.pdf Let $X$ be an affine nonsingular scheme of finite type ...
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1answer
54 views

Clifford Theorem as an easy corollary of Riemann-Roch Theorem

I'm studying Fulton's algebraic curves book and on page 109 he proves the Clifford's theorem: I have these doubts: 1.Why does he consider only the divisors $D\ge 0$ and $W-D\ge 0$? 2.What ...
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0answers
48 views

Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...
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vote
2answers
122 views

Is a specific ring extension $B$ of $K[x,y]$ integrally closed? separable?

Let $A=K[x,y] \subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
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votes
1answer
37 views

Nullstellensatz for Coordinate Ring

I'm trying to prove that if E is an irreducible algebraic set, and given any ideal A $\in$ $k[x_1,..,x_n]/I(E)$, defining V(A) = {x $\in$ E st f(x)=0 $\forall$ f $\in$ A, one has {f $\in$ ...
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votes
0answers
19 views

Dimension of curves admitting a $g_d^r$ in $\mathcal{M}_g$

Let $\mathcal{M}_g$ be the moduli space of genus $g$ curves and $\mathcal{M}_{g,d}^r = \{[C] \in \mathcal{M}_g| \text{ C carries a } g_d^r \}$ the locus of genus $g$ curves carrying a linear system of ...
0
votes
1answer
53 views

Zero set of polynomial and loops

Let $p(z)=\Re(p)+i \Im(p)$ be a non-constant (analytic) polynomial and consider the algebraic sets $Z_1=\{z\in\mathbb C:\Re(p)=0\}$ and $Z_2=\{z\in\mathbb C:\Im(p)=0\}$. Can $Z_1\cup Z_2$ contain ...
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52 views

Geometric statement of Prime Avoidance?

The Prime Avoidance Theorem is very clean to state in algebraic terms: Let $I \subset R$ be an ideal (with $R$ noetherian) and $I \subseteq \bigcup_{i=1}^r P_i$, where each $P_i$ is prime. Then $I ...
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1answer
36 views

Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like.

Let $S=\bigoplus_{n=0}^\infty S_n$ be a graded ring. We denote $S_+=\bigoplus_{n>0}^\infty S_n$. As usual we define $\text{Proj}(S)$ to be the set of homogeneous, prime ideals $\mathfrak p$ of $S$ ...
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votes
2answers
69 views

Proving that certain subset of the Grassmannian is open in the Zariski topology.

Let $\mathbb{G}(k,n)$ be the Grassmannian of $k$-planes in $\mathbb{P}^{n}$, and let $X\subseteq\mathbb{P}^{n}$ be an irreducible algebraic variety. Fix a positive integer $m$. We define $$ ...
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0answers
67 views

sections of higher direct image sheaf

Let $f:X \rightarrow Y$, be a proper birational morphism of projective algebraic varieties with $X$ smooth. Denote by $R^if_* \mathcal{O}_X$, the higher direct image sheaves. Do exists a simple way ...
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0answers
26 views

Looking for a formula to map a 2d pixel coordinate to a region within a grid.

I am given a pixel bounding box of the form: (x1, y1), (x2, y2) Where (x1, y1) is the bottom left coordinate and (x2, y2) is the top right coordinate of the ...
2
votes
2answers
94 views

When does the regularity of $A$ implies the regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
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0answers
17 views

Points of Weil restriction

Let $l/k$ be a finite separable extension of fields. Let $X$ be an $k$-scheme such that the Weil restriction $Y:= R_l/k(X_l)$ exists, where $X_l$ is the base-change of $X$ to $l$. By definition of the ...
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votes
0answers
32 views

Hartshorne Exercise 2.6: what gradation does $S(Y)_{x_i}$ inherit from $S$?

Let $S=k[x_0,\dots,x_n]$ be the "homogeneous polynomial ring" of $\mathbb P^n$ and let $S(Y)_{x_i}$ denote the localization at the image of $x_i\in S(Y)$ of $S(Y)=S/I(Y)$. In Hartshorne, he asks us to ...
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1answer
19 views

Linear system which gives $(m,n)$-polarization?

What is the dimension of $H^0(T,\mathcal{L})$, where $T$ is a complex torus of dimension $2$ and $\mathcal{L}$ is a line bundle which gives $T$ a $(m,n)$-polarization?
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1answer
35 views

A curve of genus $g\geq 2$ has a closed point of degree at most $2g-2$ over base field.

I am working on the following problem [R. Vakil] Exercise 19.8.B: Suppose $C$ is a curve of genus $g>1$ over a field $k$ that is not algebraically closed. Show that $C$ has a closed point ...
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0answers
36 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 ...
2
votes
1answer
41 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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votes
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