# 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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### dimension of linear system and multiplicity at a point

I recently encountered the following statement which I am unable to prove. Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ ...
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### showing a plane curve non-generic by exhibiting an even divisor

Let $G$ be a projective plane degree $2d$ curve with equation $AC-B^2$, with $A$, $B$, $C$ of degrees $\deg A=2d-4$, $\deg B=d$, $\deg C=4$. Then, for $d>2$, a dimension count shows that $G$ is not ...
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### What is the étale fundamental group of projective spaces over finite fields?

Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?
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### Pencil of divisors in algebraic geometry

Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and $C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics, curves or lines lying in a surface,...) ...
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### Degree of a variety vs degree of its blow-up

Let $V\subset \mathbf{P}^n$ be a variety. Let $f:\mathbb{P}^n\times \mathbb{P}^{n-1}\to \mathbb{P}^n$ be a blow-up of a point $P$ on $V$. Write $\widetilde{V}$ for the strict transform of $V$ under $f$...
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Let $X=P^{k_1}\times \ldots \times P^{k_r}$ be the product of several complex projective spaces ($P^k$ is the projectivization of $\mathbb{C}^{k+1}$) in the Segre embedding into $P=P^{(k_1+1)\cdots(... 2answers 267 views ### When is a monomial rational map on the projective space birational? Let$k$be an algebraically closed field of characteristic$0$. For$\alpha :=(a_1,\dots,a_{n+1})\in \mathbb N^{n+1}_{\ge 0}$, let$\bar x^{\alpha}:= x_1^{a_1} \dots x_{n+1}^{a_{n+1}} \in k[x_1,\...
Now to specifics: Let $V \subset \mathbb{A}^3$ be a reducible affine algebraic set defined by two irreducible polynomials $f,g \in K[X,Y, Z]$ of degree $d$ ($K$ algebraically closed). So, if $V$ is ...