Questions tagged [ag.algebraic-geometry]

Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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3
votes
0answers
210 views

Rational points on surfaces

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $...
4
votes
3answers
566 views

Deequivariantisation of indecomposable sheaves

Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
8
votes
1answer
237 views

Smooth surfaces with defective secant variety

I am interested in smooth nonedegenerate surfaces $X\subset\mathbb{P}^n$, $n\geq 5$, whose secant variety $\sigma(X)$ has dimension $4$. Clearly, the second Veronese embedding of $\mathbb{P}^2$ is ...
3
votes
1answer
147 views

Symbolic powers of a prime ideal of height one

Can someone please give me an example of a Noetherian normal local domain of dimension two such that there exists a prime ideal $P$ of height one having the property $P^{(n)}$ is not a principal ideal ...
1
vote
0answers
132 views

Symplectic structure on moduli space of holomorphic Abelian differentials

I've heard a "symplectic structure" referred to on the moduli space of holomorphic Abelian differentials by numerous people / sources. I do not know how to interpret this - I am looking for ...
4
votes
1answer
216 views

Intersection of curves in non-singular projective algebraic surfaces

Bezout thereom that says that two irreducible algebraic curves $C$ and $D$ in $\mathbb{P}^2_\mathbb{C}$ intersect at $nm$ points (counted with multiplicity), where $n$ and $m$ are the degrees of $C$ ...
1
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0answers
210 views

Questions about Hironaka's example

In Hartshorne's book 《Algebraic geomery》 p.443, the author gives an explanation of Hironaka's example on non-Kähler deformation of compact Kähler manifolds, his construction can be summarised as ...
1
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0answers
178 views

Does a ring homomorphism induce a morphism in local cohomology?

Let $\rho:R\longrightarrow S$ be a homomorphism of Noetherian rings and, for the ideals $I\subset R$ and $J\subset S$, we have $\rho(I)\subseteq J$. Does this induce a morphism in local cohomology ...
3
votes
1answer
353 views

Cohomological base change

$\require{AMScd}$ Consider the Cartesian diagram of Noetherian schemes and commutative rings $R$, $R'$: \begin{CD} N' @>{h'}>> N\\@VV{g'}V @VVgV\\ M' @>h>> M \\ @VVV @VVV \\ \mathbf{...
3
votes
0answers
138 views

What do people call functionals on holomorphic functions and on polynomials?

There are four most important functional spaces in analysis: the space $\mathcal{C}(M)$ of continuous functions on a topological space, the space $\mathcal{E}(M)$ of smooth functions on a smooth ...
2
votes
0answers
183 views

Cohomology of Beauville–Mukai varieties

The rational second cohomology of the Hilbert scheme on a K3 surface $S$ are spanned by $H^2(S,\mathbb{Q})$ plus the class of the exceptional divisor. The mapping $H^2(S, \mathbb{Q}) \to H^2(\mathrm{...
3
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0answers
130 views

Toric degeneration of Kummer Surface

I am wondering if there are any explicit examples of a toric degeneration of a Kummer surface (e.g. as a family of projective varieties say), and what the central fibre can look like? (I am working ...
8
votes
1answer
303 views

Looking for 1986 book "Algebraic and topological theories"

I am looking for the proceedings book (in paper or electronic form): Algebraic and topological theories, Papers from the symposium dedicated to the memory of Dr. Takehiko Miyata held in Kinosaki, ...
5
votes
2answers
214 views

Restricting maps between strict henselisations

$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
6
votes
1answer
145 views

$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve

Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
1
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0answers
142 views

Deformation of the trivial line bundle

Let $(\mathcal{X},\mathcal L)$ be a deformation over a (smooth) base $B$ of the pair $(X,\mathcal O_X)$ where $X$ is a smooth projective variety (over $\mathbb C$). Is the class $c_1(\mathcal L_b)\in \...
11
votes
2answers
685 views

Algebraic topology over fields other than ${\bf R}$

Is there an algebraic topology for spaces defined on fields other than ${\bf R}$, including totally discontinuous fields ? By this, I am not talking about the field of coefficients, but about the ...
10
votes
2answers
2k views

Why doesn't local cohomology seem to be used as much in algebraic geometry?

In algebraic topology, relative (co)homology is very useful. For example, we have a long exact sequence which is often helpful for lots of calculations. In algebraic geometry, we have local cohomology,...
4
votes
1answer
284 views

A reference for Serre duality

I would like some references for Serre duality for a proper noetherian scheme $X$ (over $\mathbb C$). I know that in this case there exists a dualizing sheaf. But can theorem 7.6, III, in Harshorne's &...
3
votes
2answers
275 views

When are two resolutions of a coherent sheaf homotopic

Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves). Are two ...
1
vote
0answers
86 views

Degree of a regular function on an algebraic group

Let $k$ be a field and $G$ be an affine algebraic group. We assume that $G$ is a $k$-subvariety of $\operatorname{GL}(V) \subset \operatorname{End}(V)$ for appropriate vector space $V$ of dimension $d=...
2
votes
1answer
139 views

Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties always nondegenerate

Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$. Assuming $X$ is nondegenerate and ...
3
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0answers
243 views

Most general form of Poincaré duality in étale cohomology

I am interested in Poincaré duality from the point of view of Grothendieck's 6-functor formalism. I am predominantly interested in the proof that Poincaré duality holds in étale cohomology from this ...
6
votes
2answers
278 views

Classifying space $\text{BU}(n)$ from the differential-geometric point of view?

The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...
0
votes
0answers
97 views

Problem in calculating the global sections of $\mathcal{O}_{\mathbb{P}^3}(d)\otimes \mathcal{I}_Z$

This is an additional question to the one I posed in Equivalence of sequences of blowups of $\mathbb{P}^3$ Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the ...
3
votes
0answers
80 views

Polytope algebra and toric vareties

Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$. For a simple polytope $P$ let us ...
3
votes
1answer
141 views

Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness ...
2
votes
1answer
172 views

Equivalence of sequences of blowups of $\mathbb{P}^3$

Let $[x_1,x_2,x_3,x_4]$ be coordinates of $\mathbb{P}^3$ and $Z\subset \mathbb{P}^3$ the subscheme given by the ideal $$I_Z=(x_1,x_2,x_3^2) \subset \mathbb{C}[x_1,x_2,x_3,x_4]$$ i.e. $Z$ is a double ...
3
votes
0answers
84 views

Liouville property in the Bost theorem on foliations

Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
10
votes
2answers
532 views

Reference for combinatorics with view towards representation theory/algebraic geometry

I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
4
votes
0answers
66 views

Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$

Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
1
vote
0answers
67 views

Discrepancies and multiplicity of rational singularity

Let $(X,x)$ be a rational normal surface singularity having multiplicity $m$ (for example $(-Z)^{2}$, where $Z$ is the fundamental cycle). Suppose its discrepancies are all $\ge -1+\frac{1}{k}$ for a $...
1
vote
0answers
82 views

Projectivization of normal bundle of a smooth variety in projective space

Let $X \subset \mathbb P^N$ be a smooth variety. Using Grothendieck notation, it is well-known that the conormal variety $$ \mathbb P(N_{X|\mathbb P^N}(-1)) $$ has two projections: $p_1$ to $X$ and $...
4
votes
3answers
273 views

If $X$ is a smooth $G$-variety with trivial canonical bundle, then does $X^G$ also have trivial canonical bundle?

Let $G$ be a reductive group and $X$ a smooth $G$-variety. Then the fixed point subvariety $X^G$ is also smooth (this is theorem 13.1 of Milne's book on algebraic groups). Suppose in addition that the ...
2
votes
0answers
145 views

Can components vanish without a trace?

Let $H_{P,n}$ be the Hilbert scheme of subschemes of $\mathbb{P}^n(\mathbb{C})$ with Hilbert polynomial $P\in\mathbb{Q}[t]$, and let $U_{P,n}\to H_{P,n}$ be the flat universal family. Are there $n,P$ ...
3
votes
0answers
179 views

Mixed volumes of Newton–Okounkov bodies

Let $X$ be a smooth irreducible projective complex variety of dimension $n$. Let $X=Y_0\supseteq\cdots\supseteq Y_n$ be an admissible flag. Consider $n$ line bundles $L_1,\ldots,L_n$ on $X$. Let $\...
2
votes
1answer
106 views

Extending the domain of the yoneda embedding map from étale schemes to the small étale topos so that it is still fully faithful

Let $X$ be a scheme. For $Y$ a scheme over $X$, the representable presheaf $h_Y : U\mapsto \mathrm{Hom}_X(U,Y)$ on the small étale site $X_{et}$ is actually a sheaf, and by the Yoneda lemma the ...
4
votes
2answers
543 views

Are Du Val singularities smoothable?

I am interested in when a Du Val surface singularity is smoothable. By Du Val singularity, I mean (the germ of a) isolated double point surface singularity admitting a resolution by blowups of ...
1
vote
0answers
66 views

Is the resolution of a sub-sheaf into complex of holomorphic vector bundles a "sub-resolution"?

Let $F\rightarrow X$ be a coherent sheaf on a projective Kahler manifold. We can resolve it into a complex of holomorphic vector bundles $E^{\bullet}\rightarrow X$. Let $G\subset F$ be a subsheaf of $...
3
votes
1answer
157 views

Explicit defining equations for del Pezzo surfaces

Are there known explicit parametrizations of smooth del Pezzo surfaces, say of degree $5$ or higher, as surfaces cut out by equations in some projective space? The closest I've been able to find is on ...
9
votes
1answer
565 views

"Nice" way to compute the signature of a toric manifold?

Is there a "nice" way to compute the signature of a smooth toric manifold of even complex dimension in terms of the moment polytope? By signature I mean in the sense of topology (see https://...
2
votes
1answer
117 views

Mori fiber space contractions of the blow-up of a projective space

Let $\mathbb P(V)$ be a projective space containing $Y$ as a subvariety. Let $Z$ be the blow-up of $\mathbb P(V)$ along $Y$. Clearly there exists a divisorial contraction given by $b: Z \to \mathbb P(...
10
votes
1answer
406 views

What is a "non-trivial" example of a commutative algebraic group over $\mathbb{C}$?

Let $G$ be a commutative connected algebraic group over $\mathbb{C}$. A theorem of Serre says that there exists an exact sequence $$1\to \mathbb{G}_a^n\times \mathbb{G}_m^m\to G\to A\to 1,$$ where $A$ ...
7
votes
1answer
477 views

L-functions and Galois representations: What’s the explicit relation?

It was mentioned in a lecture on Faltings’s proof of Mordell Conjecture that there’s some kind of correspondence between Galois representation (of cohomology, or some complex of constructible sheaves?)...
6
votes
0answers
275 views

Examples of descent in basic algebraic geometry

I'm studying descent theory and I recall that there were multiple instances before where I heard something like "we can prove this as follows, but this is just descent applied to [...]". ...
2
votes
1answer
166 views

Alternative definitions of étale and formally unramified in Wraith

I have stumbled upon the following definitions in a paper by Gavin Wraith. Definition 1. Say a ring morphism $A\overset \varphi \to B$ is formally unramified if every $b\in B$ admits: $b_0\in B$ ...
1
vote
0answers
74 views

Semipositive curvature on holomorphic line bundle

Let $(X,\omega)$ be a (possibly non-Kähler) compact hermitian manifold and let $L\rightarrow X$ be a holomorphic line bundle. Is there an algebraic characterization of (Griffiths) semi-positivity of $...
1
vote
0answers
116 views

Does the blow-up preserve symplectic structure?

Let $X \subset \mathbb P(\mathfrak g)$ be an adjoint variety for a simple complex Lie algebra $\mathfrak g$ appearing in the last line of the Freudenthal Magic Square, that is $\mathfrak g \in \{F_4,...
11
votes
1answer
385 views

Do smooth projective maps to $\mathbb{P}^1$ in positive characteristic have sections?

In response to the affirmative answer to this question using symplectic methods, I am wondering if the equivalent statement holds in positive characteristic? Explicitly, over an algebraically closed ...
5
votes
1answer
242 views

Square root of a line bundle up to a finite surjective morphism

Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$. The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...