# Questions tagged [projective-varieties]

In algebraic geometry, a projective variety over an algebraically closed field $k$ is a subset of some projective $n$-space $\mathbb P^n$ over $k$ that is the zero-locus of some finite family of homogeneous polynomials of $n + 1$ variables with coefficients in $k$, that generate a prime ideal, the defining ideal of the variety

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### Projectively normal and normal when the ring satisfies Serre's criterion

In Hartshorne, Ex II 5.14, says that A closed subscheme $X \subseteq \mathbb{P}_{A}^r$ is projectively normal for the given embedding, if its homogeneous coordinate ring $S(X)$ is an integrally ...
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### Vector bundles on blowups

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\ker{ker}$Let $X$ be a (projective) variety, which is singular at a point $p$ of $X$ (we can also replace $p$ with some closed subvariety). Suppose $Y$...
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### Rational normal curve as determinantal variety

We work here over complex numbers. Let $\Omega(Z)$ \begin{pmatrix} L_1 & L_2 & ... & L_n \\ M_1 & M_2 & ... & M_n\\ \end{pmatrix} be a $2 \times n$ matrix of homogeneous ...
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### Tautological bundle and its dual

Let $X=\mathbb G(2,V)$ be the Grassmannian of $2$-planes in $V=\mathbb C^n$. We denote by $\mathcal S$ the tautological bundle on $X$. In a paper there is written that "since $\mathcal S$ is a ...
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### Moduli spaces of horizontal curves

Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$. I would like to ask ...
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### Tangent bundle for orthogonal and isotropic Grassmannians

We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as $$\mathbb G(k,V):=\{W \subset V : \dim W=k\}.$$ Then consider a non-...
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### Question on a constructive proof that space projective curves are the intersections of three hypersurfaces

$\newcommand\P{\mathbb P} \newcommand\C{\mathcal C}$I am a bit confused by a proof I am reading on the fact that a projective algebraic space-curve (i.e. an algebraic curve in $\P^3(k)$, where $k$ is ...
This is a reference request/nomenclature question. Let $A \subseteq \mathbb{P}^n$ be a finite set of points not contained in a hyperplane (over some field), and let $\sigma_r(A)$ be the $r$-th secant ...