## New answers tagged ag.algebraic-geometry

2
votes

### Quotients of schemes by connected groups

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$It seems that J. Kollár gave some examples in his paper "Non-quasi-projective moduli spaces".
In pg. 1080, I think that there is an ...

5
votes

### Normalizer of solvable linear group is an algebraic group?

The normalizer of every connected closed subgroup of $\mathrm{GL}_n(\mathbf{R})$ is the stabilizer of its Lie algebra, so is Zariski-closed.

7
votes

### Question regarding the definition of linearization of line bundles

I think that you should regard the first definition as an imprecise version of the second definition. For example, suppose that $ X $ is a point and so $ L $ is simply a 1-dimensional vector space. ...

7
votes

### Frobenius and regular scheme

This is true Zariski-locally by Popescu's desingularisation theorem [Tag 07GC]. Indeed, any regular $\mathbf F_p$-algebra $A$ is geometrically regular [Tag 0381], so by Popescu's theorem it can be ...

2
votes

### Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

I will answer even though this question was asked a year ago. Hopefully my answer is of some help.
Yeah that's right.
I don't quite understand the question, the Remark 2.7 says that any transverse ...

0
votes

### Birational morphism that is not successive blow-down along smooth centers?

Let $C$ be a general curve of genus $g$ and take $\pi: Sym^g(C) \to Jac(C)$.
Then brill-noether says that that fibers of dimension $r$ arise from $h^0=r+1$ line bundles which form a $g-h^0 h^1$ ...

1
vote

### Morphisms of affine (embedded) varieties = morphisms as quasi-projective varieties?

Let $f: V \to W$ be a morphism of affine varieties (considered as quasiprojective varieties) according to the second definition. This means we have embedded $V$ and $W$ in the chart $\mathbb{A}_n \...

2
votes

### How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

Paragraph 3.2 explains that duality still works for quasi-projective non-singular varieties if you restrict to the appropriate categories. Also I think there should be no particular problem with a ...

1
vote

Accepted

### Trace morphism for projective morphism on differentials forms

For reasonable schemes, there is always a sheaf $\omega_{X/Y}$ and a canonical isomorphism
$$
\int_f \colon R^df_*\omega_{X/Y}\rightarrow \mathcal O_Y
$$
See the paper by Kleiman,
Relative duality ...

3
votes

### A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?

Here a maybe not too important problem for a commutative algebra ,which is solved using non-commutative algebra and the solution is quite non-trivial:
Problem: Let $K$ be a field, classify the finite ...

Community wiki

-4
votes

Accepted

### A problem in commutative algebra whose solution requires algebraic geometry (resp., noncommutative algebra)?

The deficit of any clear, good, and important examples leads me to believe that there might not be any known examples. Commutative rings are to schemes as perfectoid algebras are to perfectoid spaces,...

Community wiki

0
votes

### algebraic de Rham cohomology - open subvariety and normal crossing

See Theorem 5.9 of the following article:
https://impa.br/wp-content/uploads/2022/01/33CBM17-eBook.pdf.
Bott-Tu relative algebraic de Rham cohomology is what Friedlich has studied. I didn' check ...

1
vote

Accepted

### Zero loci of sections of wedge product of bundles

A global section of $\wedge^2F$ is a bivector $\xi \in \wedge^2V$
and the zero locus of such a section is the scheme parameterizing all 2-dimensional subspaces $U \subset V$ such that the image of $\...

22
votes

Accepted

### Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points

A counterexample is given by the following five points:
$$(0,0),(1,0),
\Big(-\frac{64867}{77629},\frac{3389}{60094}\Big),
\Big(\frac{5981}{56176},\frac{32211}{34172}\Big),
\Big(\frac{5925}{117812},-\...

8
votes

Accepted

### Three-dimensional analogues of Hirzebruch surfaces

The 3-dimensional analogues should be $\mathbb{P}(O(a)\oplus O(b)\oplus O(c))$, yes. These are exactly the $\mathbb{P}^2$-bundles over $\mathbb{P}^1$.
In general, any projectivization of a vector ...

2
votes

Accepted

### Derived algebraic geometry and Bridgeland stability conditions

Your question was already asked and answered. Below is what I wrote to answer the linked question.
This is chapter 7 of Fosco Loregian's thesis, linked from his webpage. The paper Simone Virili linked ...

4
votes

Accepted

### Question about a remark on quantization of Coulomb branches

The algebra of difference operators is very explicitly described at the beggining of the appendix (section A(i)) of Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine ...

0
votes

### The pseudoeffective cone does not contain lines

For a smooth projective variety, this follows from Lemma 2.3 in https://arxiv.org/abs/1206.6521.

5
votes

Accepted

### An example of a geometrically simply connected variety with infinite Brauer group (modulo constants)

This is an open problem; I personally suspect it cannot happen.
More generally let $X$ be a smooth projective variety over a field $k$ which is finitely generated over $\mathbb{Q}$. Then Skorobogatov ...

5
votes

Accepted

### Derive distributional inequalities from pointwise estimates

First note that the result is true if the support of $\varphi$ is disjoint from $E$; this follows from integration by parts.
Next note that every algebraic set is a finite union of smooth submanifolds ...

6
votes

### Negative intersection number between curve and effective divisor

If $X$ is a smooth projective variety of dimension $d$ and $\tilde X \to X$ is the blowup of $X$ at a point $x$, then the exceptional divisor $E \subseteq \tilde X$ is isomorphic to $\mathbf P^{d-1}$ ...

5
votes

### Negative intersection number between curve and effective divisor

It is not. Let $X$ be a smooth surface and let $X^{[n]}$ be the Hilbert scheme of $n$ points on $X$, parameterizing zero-dimensional subschemes of $X$ of length $n$ . This is a smooth complex ...

2
votes

### Why is "everything staying correct" for simplicial spaces?

In general, it is not true that in the context you are interested in, simplicialization of statements preserves their truth. For example, already “every epimorphism of sets is a retraction”, but not “...

1
vote

Accepted

### Some folklore about crystaline rings of differential operators

Proposition 1* does follow from that $\mathcal D_c$ sheafifies. For if $k(X) \cong k(Y)$, then $X$ and $Y$ are birational, so there are affine open subsets $U \subseteq X$, $V \subseteq Y$ such that $...

6
votes

### $n$-th root of meromorphic functions of several complex variables

If $\Omega$ is simply connected, this is true.
Let $\tilde \Omega$ be the space of pairs $(z, g)$ where $z \in \Omega$ and $g$ is the germ of an $n$th root of $f$ at $z$. If $\mathrm{div}(f) = nD$, ...

1
vote

Accepted

### A graphic representation of classical unitals on 28 points

Using the answer of Taras Banakh, I have implemented an interactive example using HTML and Javascript. It shows all lines of the unital with random colors. Every line is clickable. After clicking on a ...

3
votes

### For an element in the integral closure of an ideal $I$ - which power is in $I$?

Not a complete answer to your question, but the number $k_0$ (or $k_0-1$) is called the Briançon-Skoda number of $R$. For analytic planar curves, there is a formula for this number in
J. Sznajdman: ...

4
votes

Accepted

### Irreducibility of an explicit complex projective variety

Let me explain how to show that the projective surface $\Sigma$ is geometrically irreducible (see also the comments above).
First, we know that $\Sigma$ is irreducible (this was checked by the OP).
...

4
votes

### Does there exist a faithful exact embedding of $D^b(\dim(N)) \to D^b(\dim(N-1))$

If you're not assuming full you can use Prop. 2.3 in
Canonaco, Alberto; Stellari, Paolo, Non-uniqueness of Fourier-Mukai kernels, Math. Z. 272, No. 1-2, 577-588 (2012). ZBL1282.14033.
to prove the ...

2
votes

### Does there exist a faithful exact embedding of $D^b(\dim(N)) \to D^b(\dim(N-1))$

This is answered by Noah Olander in Fully Faithful Functors and Dimension.

1
vote

### Example for simply connected variety with trivial holomorphic Euler characteristic

I don't have complete confidence in this answer, but: Table $3$ in Iano-Fletcher lists parameters $(a_0, a_1, a_2, a_3, a_4)$ such that a general hypersurface of degree $1+\sum a_i$ in the weighted ...

0
votes

### Equivalence of dihedral and symmetric group actions on a specialized real algebra

Equivalence of Group Actions $D_3$ and $S_3$ on Floretion Algebra $A(F_n)$
I propose a visual and algorithmic approach to demonstrate the equivalence of two group actions on the algebra over the ...

11
votes

Accepted

### Intersection cohomology and Poincaré duality

Let $E$ be an elliptic curve and let $Y$ be the union of two $\mathbb P^1$s joined by a node (say, the solution set of $xy = 0$ in $\mathbb Z/2$).
Let $\sigma$ be an involution of $E \times Y$ that ...

4
votes

### Grassmannian $\mathrm{Gr}(k, \pm \infty)$ in infinite dimension

You should have a look at the Appendix to this paper of mine:
https://arxiv.org/abs/1212.3528
https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/jdt064

8
votes

### Homogeneous polynomials cutting out complex abelian varieties

Horrocks-Mumford surfaces are cut out in ${\mathbb P}^4$ by 3 quintic and 15 sextic polynomials; the equations will have many dependencies (syzygies) between them. The references I found are [...

4
votes

Accepted

### Homomorphism between ideal sheaves of codimension $2$?

The main observation is that, if $Y \subset X$ and $\operatorname{codim}(Y) \ge 2$ then
$$
(I_Y)^\vee \cong \mathcal{O}_X,
\qquad\text{hence}\qquad
(I_Y)^{\vee\vee} \cong \mathcal{O}_X.
$$
Now assume $...

4
votes

Accepted

### Deligne-Lustzig varieties locally closed schemes

Any set which is open in its topological closure can be endowed with a reduced scheme structure in the way you disguise. The set will also be the intersection of an open set and a closed set, so you ...

2
votes

### Number of Plücker relations for a Grassmannian

The total number of Plücker relations is not the relevant question, since they are not independent; algebraic relations hold between them. In fact on various dense (Zariski) open sets, defined by the ...

4
votes

### Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

There are likely more-elementary approaches, but here is an argument using some basic function-field arithmetic. Let's work a bit more generally than in the original question, considering a base field ...

3
votes

### A graphic representation of classical unitals on 28 points

Now I have understood how to draw the classical unital, $U_q$ at least for $q=3$. It is known that this unital is isomorphic to the subset of the projective plane $PG(2,9)$, defined by the equation $X^...

2
votes

### Conceptual explanation for extra/missing $p$ solutions to $x^2+y^2=a \pmod p$ at $a=0$

I'll begin with a disclaimer that this response doesn't actually resolve the numerical coincidence; I don't provide any direct argument that points gained must equal points lost. But I do show a ...

4
votes

### General algebraic result obtained from consideration on $\mathbb{Q}_p$

Matt Baker gave a proof that $(\mathbf Z/p\mathbf Z)^\times$ is cyclic (always a bit mysterious, as there's no "natural" generator) by first looking at a related result in characteristic $0$ ...

8
votes

Accepted

### Exactness of the Weil restriction functor $\mathrm{Res}_{X/k}$

It is not right exact. Assume that $k$ is algebraically closed. If the map $Res_{X/k}B\to Res_{X/k}C$ was surjective as a map of sheaves for the fppf topology, then in particular, the map on sections $...

2
votes

Accepted

### How to complete $f^*f_*G\to G$ and $F\to f_*f^*F$ into a distinguished triangles for a double branched covering $f:X\to M$?

The main question can be answered as follows. Note that $f_*$ and $f^*$ are both Fourier-Mukai functors with the FM kernels given by the structure sheaf of the graph
$$
\mathcal{O}_{\Gamma(f)} \in \...

1
vote

Accepted

### What are the finite étale coverings of a quasi-hyperelliptic surface?

From pp. 488-489 of W.E. Lang's thesis (Ann. ENS. vol. 12, 1979) it follows that neither case can arise. For the fibration $X\to C$ has an elliptic base and all its geometric fibers are cuspidal ...

12
votes

Accepted

### Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD

Here is a proof using a bit of commutative algebra and algebraic geometry, elaborating on comments by @abx and @ChrisWuthrich. The ring
$$R=\mathbb{C}[x,y]/(x^3+y^3-1)$$
is a Dedekind domain, because ...

3
votes

### Is there an isotrivial elliptic surface of positive rank having a section of order $3$?

Let $k$ be a field and let $n\geq 3$ be an integer which is invertible on $k$.
The following lemma will allow you to conclude that an elliptic curve over $k(t)$ which is isotrivial with $j$-invariant ...

3
votes

### A simple question about a statement of Kahler Manifold and Moishezon Manifold

The answer probably depends a lot on how exactly you define a Kähler manifold as there are multiple equivalent definitions. In any case, the definition of a positive line bundle is that it has a ...

3
votes

Accepted

### On the situation of intersections along a proper morphism

There is a general fact that a proper affine morphism is finite, and a general trick that for $\overline{X} \to S$ proper containing an open subset $X\to S$ affine, a closed subscheme $Z$ of $\...

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