# Questions tagged [ag.algebraic-geometry]

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

18,832
questions

**3**

votes

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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**

vote

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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**

vote

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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}^...