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Questions tagged [projective-schemes]

This tag is for questions relating to "projective scheme".

A projective scheme is a closed sub-scheme of a projective space $P_{\mathbb{k}}^n$. In homogeneous coordinates $x_0, \ldots, x_n$ on $P_{\mathbb{k}}^n$, a projective scheme is given by a system of homogeneous algebraic equations:

$$f_1(x_0, \ldots, x_n) = 0, \ldots, f_r(x_0, \ldots, x_n) = 0$$

371 questions
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Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or alternatively due to page 79 in this script)…
user267839
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How to complete Vakil's proof that the composition of projective morphisms are projective when the target is quasicompact?

For this question, a morphism $\pi : X \rightarrow Y$ is projective iff there exists a finite type quasicoherent sheaf $\mathcal{E}$ on $Y$ such that $X$ is isomorphic (as a $Y-$scheme) to a closed subscheme of $\mathbb{P}(\mathcal{E})$. I am…
Tom Oldfield
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Tangent space in a point and First Ext group

Let $X$ be an abelian variety over an algebraically closed field $k$. I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification $$T_x(X)\simeq \operatorname{Ext}^1(k(x),k(x))$$ where on the left we have the tangent…
Veen
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Classifying Quasi-coherent Sheaves on Projective Schemes

I know some references where I can find this, but they seem tedious. Both Hartshorne and Ueno cover this. I am wondering if there is an elegant way to describe these. If this task is too difficult in general, how about just $\mathbb{P}^n$? Thanks!
BBischof
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global sections of structure sheaf of a projective scheme X over a field k?

Some may have already asked this question. What are the global sections of structure sheaf of a projective scheme $X$ over a field $k$? By Hartshorne page 18, Chapter 1, Theorem 3.4, global sections will be $k$ when $k$ is algebraically closed and…
user48537
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Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is also a regular curve, then I think I can do this, as it…
DCT
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Is the graph of morphism of projective varieties $X \rightarrow Y$ closed in $X \times Y$?

The graph of a morphism $X \rightarrow Y$ is closed in $X \times Y$ if $X$ and $Y$ are affine varieties. What if $X$ and $Y$ are projective varieties? I am still not quite familiar with projective varieties. So I need some help. Thanks very much.
ShinyaSakai
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Proj of Almost Same Graded Rings are Isomorphic (Exercise from Vakil's FOAG)

I'm trying to solve Exercise 6.4.F from Vakil's FOAG: 6.4.F. Exᴇʀᴄɪsᴇ.$\quad$Show that if $R_\bullet$ and $S_\bullet$ are the same finitely generated graded rings except in a finite number of nonzero degrees (make this precise!), then…
user940160
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Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } S_{(f)}$, the latter denoting the degree zero…
Future
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Compute the cohomology of projective schemes

In Hartshorne's book, Section 3.5, the cohomology of projective spaces is computed. How to compute the cohomology of projective schemes? Maybe the general case is complicated, please look at the following simple case: Let $R=k[x_1,\dotsc,x_n]$ with…
Strongart
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Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is obvious intuitively if one thinks of differential forms and…
Potato
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Does a dualizing sheaf $\omega_X$ give rise to a dualizing module?

Let $X = \text{Proj } R$ be a projective equidimensional Cohen-Macaulay scheme, where $R$ is a finitely generated graded Cohen-Macaulay $\mathbb{C}$-algebra and $\mathcal{O}_X(1)$ is ample. Suppose that the induced homomorphism $R \to…
Mellon
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Classification of automorphisms of projective space

Let $k$ be a field, n a positive integer. Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo scalars. His hint is to show that $f^\star…
only
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Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field $k$, and let $\mathcal{F}$ be a coherent sheaf on…
Nadim Rustom
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Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to be $(M_{(f)})^\sim$ (where this is the associated…
Potato
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