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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.

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Geometric genus of projective bundle over a scheme?

This is Hartshorne's AG, exercise III.8.4(d): Let $Y$ be a noetherian scheme and $\mathscr{E}$ be a locally free $\mathscr{O}_Y$-module of rank $n+1$. Let $X=\mathbb{P}(\mathscr{E})=\underline{\mathrm{Proj}}_Y(\mathrm{Sym}(\mathscr{E}))$. Let…
WakeUp-X.Liu
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Duality of an Ample invertible sheaf has no non-trivial global section?

This is EXERCISE 7.1 of Chapter III in Hartshorne's AG: [Q] Let $X$ be an integral projective scheme of dimension $\geq1$ over a field $k$, and let $\mathscr{L}$ be an ample invertible sheaf on $X$. Then $H^0(X,\mathscr{L}^{-1})=0$. Actually if…
WakeUp-X.Liu
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Fixed point property for projective space

Let K be an algebraically closed field. Is it true that any endomorphism $\mathbb{P}_K^n\to \mathbb{P}_K^n$ has a fixed closed point? For $K=\mathbb{C}$ the (oriented) topological version is a classical result and the proof uses Lefschetz fixed…
Kamil
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Tangent space of quasi-projective varieties

If $X$ is a quasi-projective variety and $X_i,\;\;i=1,\ldots,k\;$ are its irreducible components, then why $$\mathrm{dim}\;T_{X,x}=\mathrm{max}_{i=1,\ldots,k}\;(\mathrm{dim}\;T_{X_i,x})?\qquad \qquad (x\in X)$$
Jacob Fox
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Regular schemes and base change

Suppose $X$ and $Y$ are regular schemes (i.e. all local rings are regular) and flat over some base $S$. Assume further that $Y$ has relative dimension $0$ over $S$. Does it follow that $X \times_S Y$ is regular? I'm really interested only in the…
regular
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Proposition 2.31 of Mumford's algebraic geometry I

I'm reading Algebraic Geometry I: Complex Projective Varieties by Mumford and have trouble understanding the proof of proposition 2.31. Let $p_2$ be a projection, and the following proof is only partial. (2.31) Proposition. Let $S \subset…
mikecat1024
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Irreducible subspace of $\mathbb{A}^2$

Let $X:=V(x^m-y^n)$ be a subspace of $\mathbb{A}^2$. How can I prove that if $(n,m)=1$ then $X$ is irreducible? I think that it is isomorphic to $\mathbb{P}^1$ but I can't prove that.
Jacob Fox
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Zariski tangent space of a point viewed as a point of a subvariety

Let $X \subset \mathbb{C}^n$ be an affine variety (not irreducible). Let $Y$ be a subvariety of $X$ (again not irreducible). How can we relate the Zariski tangent space at $P \in Y$ and at $P \in X$? (Corrected after Mariano's comments) Based on my…
Manos
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Is a product of curves a complete intersection?

Let $C_1, C_2$ be two projective smooth curves over $\mathbb{C}$. Is it possible to say when $C_1 \times C_2$ a complete intersection in some projective space? For three curves the answer is "it is never a complete intersection" since a product of…
iou
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Is numerical equivalence preserved by base change to an algebraically closed field?

Vakil's Foundations of Algebraic Geometry, Proposition 20.1.4 is that intersection multiplicity depends only on numerical equivalence classes. The proof uses base change to an algebraically closed field. I'm not sure why we can do this. Let $X$ be a…
David Lui
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How to understand the definition of the base locus and the base ideal of a complete linear system?

Let $X$ be a projective variety and $D$ a Cartier divisor on $X$. In the book Positivity in AG, the definition of the base ideal of $|D|$ is the image of the map $$eval_{|D|}: H^0(X,D) \otimes O_X(-D) \to O_X .$$ And the base locus of $|D|$ is…
Hobo
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Intersection of a hyperplane and a curve

Let us fix a projective curve X over a field k. With nt, I mean a variety with all irreducible components of dimension 1. Let us suppose that there is a smooth rational point $x \in X$. My question is: Is it possible to find a hyperplane such that…
Riemannrock
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Relation between the points of codimension 1 and the generic points, in scheme.

For a commutative ring $A$, let $R(A)$ be the set of regular elements in $A$ and $\textrm{Frac}(A)=R(A)^{-1}A$. It is known that if $A$ is Noetherian local with $\textrm{dim}(A)=1$, then $\textrm{length}(A/fA)<\infty$ for any $f\in R(A)$. Then we…
User0829
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If a divisor $D$ on a surface has positive self-intersection, then $nD$ or $-nD$ has nontrivial sections, eventually in $n$.

Let $S$ be a smooth complex projective surface. Let $D$ be a divisor on $S$, such that $D^2 = D.D> 0$. Then, at least one of the following holds: For $ n \gg 0$, $H^0(nD) \neq 0$; for $n \gg 0$, $H^0(-nD) \neq 0$. Here $H^0(D)$ denotes the space…
Francesco Genovese
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A sequence of divisors on surfaces

Let $X$ be a smooth, complex projective algebraic surface. Let $C,D$ be two nonzero effective divisors on it. Then in literature one can find the following exact sequence : $0 \to \mathcal O_D(-C) \to \mathcal O_{C+D} \to \mathcal O_C \to 0$. I'm a…
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