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Questions tagged [algebraic-geometry]

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. This tag should not be used for elementary problems which involve both algebra and geometry.

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. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
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Can Bezout's theorem be generalized to non algebraically closed fields?

The Bezout's theorem says that the intersection of two curves in $\mathbb{P}^2_k$, (counting multiplicity, $k$ is algebraically closed) is equal to the product of their degrees. Can the theorem be generalized to non algebraically closed field? Where…
user93417
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meromorphic functions on proper varieties are rational

Suppose $X$ is a proper variety over $\mathbb{C}$, is every meromorphic function rational? In the case of projective variety, can this be derived from Chow lemma? How does the GAGA principal illustrate on these ?
user93417
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Two definitions of the Weil restriction.

Let $L/K$ be a galois extension with $G:=\mathrm{Gal}(L/K)$, $X$ a $L$-scheme. We have two definitions of the Weil restriction of $X$ : 1) If the contravariant functor $\mathrm{Res}^L_K(X) : (Sch/K) \rightarrow Set, T \mapsto X(T\times_K L) $ is…
gpst
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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 subspaces of $\mathbb P^n(k)$ ($m= 0, 1, \dots n$). For…
user143145
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Identifying the rational normal curve with "squares" in $\mathbb{P}(\operatorname{Sym}^2(V))$

Suppose V is a two-dimensional vector space. According to Harris' "Algebraic Geometry", p.118 the vectors in $\mathbb{P}(\operatorname{Sym}^2(V))$ of the form $v\cdot v$ are supposed to be exactly the elements of $v_2(\mathbb{P}^1)$, which is the…
Cedric B
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question of some invertible sheaves

Let $C$ be a nonsingular curve and $\mathcal{F}$ be a locally free sheaf with rank $2$. suppose that $H^0(\mathcal{F})\neq 0$ and let $s \in H^0(\mathcal{F})$ be a nonzero section. Then $s$ determines an injective map $\mathcal{O}_C \rightarrow…
Sang Cheol Lee
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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 in $\mathbb{P}^d_{k(y_0)}$. Set $r = dim…
user101036
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If $f$ is birational, is the pushforward map on the numerical groups surjective?

Say that I have a morphism of projective algebraic varieties $f: X \to Y$, which is birational. There is a pushforward of cycles morphism $f_*: N_*(X) \to N_*(Y)$. Now, if I could pull back cycles and if I had a projection formula then I could say…
Jacob Bell
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Question of the relation between very ampleness and irreducibility

Let $X$ be a projective surface and $D$ be a divsor. Then I know $D$ correspond to a curve of $X$. My qeustion is simple. If $D$ is very ample, then the corrsponding curve of $D$ is irreducible? More generally, if $X$ be a projective variety of…
Sang Cheol Lee
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affine scheme over a ring R

I red an article and encountered some concepts from algebraic geometry. Let $R=\mathbb{Q}[\alpha_1,\ldots,\alpha_5]$ be a polynomial ring in the variables $\alpha_i$. Define $f(x,y)\in R[x,y]$ by $$f(x,y)=y^2+\alpha_1 xy+\alpha3 y-x^3-\alpha_2…
Nadori
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Pascal's theorem by Bezout's theorem

I need to prove the following theorem Let the hexagon $ABCDEF$ be inscribed in the nondegenerate conic $q=V(f)$. Assume that $A,B,C,D,E,F$ are distinct. Let $P=\overline{FA}\cap \overline{CD}, Q=\overline{AB}\cap \overline{DE}$ and…
Jimmy R
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Some technical details about regular functions on quasiprojective varieties

I have some confusion about some very basic notions in algebraic geometry. I am using Shafarevich but I find the book to be pretty unclear at times. First, given a quasi projective variety $X$ over an algebraically closed field $\mathbb{k}$…
Seth
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When divisor self-intersection number equal to 0?

Let $D$ be a divisor on smooth rational surface $X$ (over $\mathbb{C}$) such that the linear system $|nD|$ has dimension 1 (fore some $n$), has no fixed components and base points. How can I show that $D^2=0$? Thanks!
Andrew
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Why a closed subscheme give rise to a closed immersion

G\"ortz and Wedhorn, in their book ``Algebraic Geometry I'' at page 84, give the following definitions. Let $(X, \mathscr{O}_X)$ be a scheme. (1) A closed subscheme of $X$ is given by a closed subset $Z \subseteq X$ (let $i : Z \longrightarrow X$ be…
Eskil
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Flat morphism of finite type

Let $f:X \rightarrow Y$ be a finite dominant morphism of integral locally noetherian schemes. If f is flat over a point $y$, is it true that we can find an open nbhd V of Y ontaining y such that $f^{-1}(v) \rightarrow V$ is flat? I think this…
user101036
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