Questions tagged [ag.algebraic-geometry]
for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
14,707 questions
0
votes
0answers
43 views
Flatness of the integral closure
Let $R$ be a $p$-torsion free ring which is integrally closed in $R[1/p]$ and let $S$ be a finite etale extension of $R[1/p]$.
Is it true that an integral closure $S^+$of $R$ in $S$ is flat over $...
2
votes
0answers
41 views
Finding a curve through divisor
Let $k$ be a field. Is it true that for any smooth irreducible projective $k$-variety $X$ and a dense open set $U\subset X$, for any zero-cycle on $X$ one can find an irreducible curve containing its ...
1
vote
0answers
29 views
On the exponent of a certain matrix $A$ in characteristic $p > 0$
Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p + \cdots + p^i)$, where $i > 0$.
Suppose that further the $(m,n)$-component $a_{m,n}$ of the ...
-4
votes
0answers
48 views
Euclidean Geometry [on hold]
In Euclidean Geometry we can create a square with side length of $\sqrt2$. as we know, $\sqrt2$ is a number which has no end. so it is physically impossible to have a square with this length in the ...
-3
votes
0answers
48 views
Example coordinates for a dodecahedron in x, y, z? [on hold]
Good morning/afternoon/evening all,
I'm working on some high-level Python simulations and wanted an example set of coordinates for the vertices of a regular dodecahedron in terms of x, y and z ...
1
vote
0answers
88 views
Status of locality of perfectoidness for uniform rings
Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$...
0
votes
0answers
53 views
Regular intersecting family
Let $n=2k+1$. When $k=3$, the set of lines of Fano plane is a regular intersecting family consisting of some k-subsets of [n]. Do anyone know such examples for general $k$? Thanks.
3
votes
1answer
179 views
Hochschild Homology and Formal Geometry
My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$.
The spectral sequence arises from the ...
1
vote
1answer
114 views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
2
votes
0answers
91 views
Berthelot’s comparison theorem and functoriality
Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$.
Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
5
votes
1answer
144 views
Multiple mirrors phenomenon from SYZ and HMS perspective
There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
2
votes
0answers
172 views
Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?
Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
1
vote
0answers
155 views
p-adic Hodge theory for singular projective varieties
In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...
1
vote
1answer
96 views
Complete intersection subvariety of projective variety
I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space.
My guess is, a subvariety X of ...
11
votes
1answer
387 views
+500
Applications of integral p-adic Hodge theory
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...
15
votes
1answer
261 views
Where to find some subset of Khovanskii's 15 proofs of the BKK theorem?
(I asked this question on MSE, but someone suggested it would be better asked here.)
I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a ...
1
vote
0answers
81 views
Factorizations of etale morphisms
Let $f:X \rightarrow Y$ be a finitely presented separated etale morphism, with $Y$ quasicompact and quasiseparated.
By Zariski’s main theorem, we can factor $f$ as $f= g \circ j$ with $j$ an open ...
4
votes
1answer
147 views
rank of Jacobian of Fermat curve and Chabauty-Coleman method
Consider the fermat curve $F(p)$ over $\mathbb Q$ which is the projective closure of $X^p+Y^p=1$ inside projective plane, where $p$ is a prime number and without loss of generality we assume $p>2$. ...
3
votes
0answers
107 views
Quotient of a smooth projective surface by an involution
Is the quotient of a smooth complex projective surface by an involution projective? Suppose the quotient happens to be smooth; does that change the situation?
1
vote
0answers
109 views
Sources of derived schemes in geometric representation theory
What are some derived schemes naturally arising in geometric representation theory?
Some examples include:
Steinberg scheme
Hilbert scheme
Moduli stack of local systems.
Now, this looks like a ...
2
votes
1answer
271 views
A clarification of an argument in “Perfectoid spaces”
The page 50 of (the arXiv version of) the above-mentioned paper of P. Scholze says "Now the Poincare duality pairing implies that $H^i(Y_{\mathbb{C}_p, et}, \bar{\mathbb{Q}}_l)$ is a direct summand of ...
3
votes
0answers
114 views
Are Chain Complexes Related to the Tangent Bundle Construction?
For a scheme $X$ over $\text{Spec}(K)$, we can consider maps $\text{Sch}(\text{Spec}(K[d] / d^2), X)$, which we can think of as the tangent bundle over $X$. A map $\text{Spec}(K) \rightarrow S$ picks ...
1
vote
1answer
82 views
Linear operator on polynomials and invariant sets of roots
Let $T:\mathbb{C}_n[x] \to \mathbb{C}_n[x]$
be a linear map from the vector space of polynomials of degree $n$ to itself.
Let $S \subset \mathbb{C}$ be a set with at least $3$ points, such that
for ...
7
votes
2answers
125 views
The degree of the hypersurface of pfaffian cubic fourfolds
Let $\Pi:=\mathbb{P}(H^0(\mathbb{P}^5,\mathcal{O}_{\mathbb{P}}(3)))$ be the space of cubic fourfolds in $\mathbb{P}^5$. It is well-known that those cubics which are pfaffian, i.e. defined by the ...
4
votes
1answer
133 views
Morphism of sites and abelian sheaf cohomology
Let $f : \mathcal{C}\to\mathcal{D}$ be a morphism of sites (see the Stacks Project) with induced morphism of topoi
$$(f^{-1}, f_*) : Sh(\mathcal{D})\to Sh(\mathcal{C}).$$
By assumption, $f^{-1}$ is ...
2
votes
0answers
130 views
Schemes obtained replacing variables by $n \times n$ matrices?
Let $k$ be an algebraically closed field and $n \geq 1$ be an integer. Choose a polynomial $f \in k[x_1,\dots,x_m]$ and place the variables in a fixed order (i.e we choose a preimage of $f$ inside $k\...
5
votes
0answers
309 views
+50
Are nearby points in an algebraic curve necessarily connected?
I would like a result of the following form:
For every algebraic curve $C$ in $\mathbb{R}\mathbf{P}^{n-1}$, there exists an
explicit and easy-to-compute $\epsilon=\epsilon(C)>0$ such that ...
2
votes
2answers
112 views
Construction of Fano threefold of degree $5$ and its defining equations
The Fano threefold $X$ of index $2$, degree $5$ and Picard number $1$ is known to be a general codimension $3$ linear section of the $Pl\ddot{u}cker$ embedding of Gr(2,5).
My first question: what ...
4
votes
0answers
147 views
Regular functions vs holomorphic functions
Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...
-7
votes
0answers
133 views
Grothendieck, Sphere, Cylinder [closed]
Archimedes' theorem relating the surface areas and volumes of the sphere and cylinder is well known. I'm wondering if scheme theory (specifically etale cohomology of schemes) may be useful to gain ...
13
votes
1answer
481 views
GAGA for henselian schemes
In this paper, F. Kato recollects basic facts on henselian schemes and proves some partial results towards GAGA in the context of henselian schemes.
Let $I$ be a finitely generated ideal in a ...
2
votes
0answers
82 views
Linear projection from a point preserves flatness
Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
3
votes
0answers
76 views
Degeneration of cycle class map
Let $f:\mathcal{X} \to \Delta$ be a flat family of projective varieties, smooth over the punctured disc $\Delta^*$ and the central fiber is a simple normal crossings divisor. Let $\mathcal{Z} \subset \...
1
vote
0answers
68 views
Minimal size of an open affine cover for an open complement
Let $X$ be a smooth projective scheme and $Y$ be a projective subscheme of $X$, not necessarily smooth. Are there any known results about the minimal size of an open affine cover (number of affines in ...
6
votes
0answers
136 views
An inverse problem for Grothendieck rings of varieties
Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...
7
votes
1answer
212 views
Deligne Pairing v.s. Weil Pairing on a Family of curves
We have the Deligne Pairing on a family of curve $\pi:X\to S$ by using
$$\langle L,M\rangle_{\mathrm{Pic}^0(X/S)}=\det R\pi_*(L\otimes M) \otimes (\det R\pi_*L)^{-1}\otimes (\det R\pi_*M)^{-1} \otimes ...
1
vote
0answers
48 views
Computing Gröbner basis elements of some constant degree
I'm wondering if there is any way or any special set of ideals such that there is an efficient way to compute elements of degree at most $d$ in a Gröbner basis for that ideal.
If you have any paper ...
3
votes
2answers
211 views
What do non-principal divisors in a Picard group look like
The way Divisors on Elliptic Curves are motivated in cryptography is to say it's a convenient way to represent rational functions by keeping track of multiplicities of the zeros and poles of a ...
1
vote
0answers
91 views
Birational maps mapping ample class to ample class?
I refer to the paper "Normal Subgroups in the Cremona Group". In the last paragraph of the proof of proposition 5.13, the author wrote the following: "Assume now that $h\in \text{Bir}(X)$ preserves ...
2
votes
0answers
178 views
Geometric genus after projection
I am using the software "Singular" for computing the geometric genus of a space curve, but something is going wrong. Let me consider the following
3-dimensional curve:
$$\begin{array}{c}x_3 D(x_1,x_2)...
4
votes
0answers
120 views
Local structure of non-normal toric varieties---possible mistake in “Discriminants, Resultants and Multidimensional Determinants”
I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...
0
votes
0answers
14 views
Parametric research for the global minimum in a family of polynomial multivariable functions on closed domains
Consider the family of functions:
$$
V(\{x_j\},\{y_j\})=\sum_{j=1}^L\left[\frac{1}{2} x_j^2+ \frac{\beta^2}{2}y_j^2 + \alpha\beta\, x_jy_j \right]
$$
Each member of the family is therefore ...
6
votes
1answer
162 views
Vanishing of higher direct image of finite morphisms relative to the fppf topology
Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
4
votes
0answers
107 views
Poincare duality in families of smooth, projective curves
Let $f:\mathcal{C} \to \Delta^*$ be a family of smooth, projective curves over a punctured disc. Denote by $\mathbb{H}^1:=R^1f_*\mathbb{Z}$ the associated local system, such that for every $t \in \...
4
votes
0answers
137 views
Torsor descriptions of $Bun_G$
The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero ...
4
votes
0answers
120 views
Explicit Enriques involutions on the Fermat quartic surface
Let $X$ be the complex Fermat quartic surface defined by the polynomial $x^4+y^4+z^4+w^4$.
By results of Sertöz, we know that the surface $X$ admits at least one Enriques involution, i.e. an ...
5
votes
1answer
257 views
Does there exist a discrete valuation subring $R$ of $K((t))$ ($K$ a number field) of residue characteristic $p$ with $\mathrm{Frac}(R) = K((t))$?
Let $K$ be a number field, and let $K((t))$ be the field of formal Laurent series. Let $p > 0$ be a prime.
I have two questions:
Does there exist a discrete valuation subring $R$ of $K((t))$ of ...
1
vote
0answers
126 views
Cohomology of constant sheaves
Let $X= spec(k)$ where $k$ is an algebraically closed field. Consider the constant sheaf $\mathbb{Z}$ on the fppf site of $X$. I'm interested in computing $H^1_{fppf}(X, \mathbb{Z})$. I know that $H^...
2
votes
0answers
65 views
Extending Normal Bundle of a Subvariety of $\mathbb P^n$
This question is related to, but not the same as, an earlier question: Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
Given a smooth projective variety $X\subset\mathbb P^n$, let $N_X$ be ...
2
votes
1answer
96 views
Resolution by locally free $G$-equivariant sheaves on varieties
I have been reading the section in the beginning of Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics by Bartocci et. al., and stumbled across the following sentence (page 26).
...
