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

learn more… | top users | synonyms (1)

1
vote
1answer
82 views

How to define the internal hom between presheaves valued in cotensored categories?

First let $\mathcal{V}$ be a closed symmetric monoidal category and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over ...
1
vote
0answers
112 views

addition on an affine scheme [closed]

At the Brenner's introduction to the geometric view of the tight closure the author states that an affine scheme has a natural addition (this addition will be extended to the vector bundles). I wonder ...
5
votes
3answers
232 views

set of centers of sphere inscribed in tetrahedron

Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced ...
1
vote
1answer
180 views

Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...
1
vote
0answers
109 views

How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$ with cone singularities along divisor $D$ with cone angle $2\pi\beta$ is said to be of constant scalar curvature Kahler or cscK if its scalar curvature $S(\omega)$, which is ...
0
votes
1answer
212 views

Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field. Is there a full ...
4
votes
0answers
230 views

Motivic fundamental group of the moduli space of curves?

Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
1
vote
1answer
113 views

Vanishing for ideal sheaves on spaces with only rational singularities

I am trying to find a more elementary proof of [GKKP-DIFFERENTIAL FORMS ON LOG CANONICAL SPACES, Corollary 13.4] for rational singularities, avoiding Du Bois pairs. When $X$ is a rational singularity ...
3
votes
0answers
214 views

Why does this example of global residue theorem not work?

This question was previously asked here. I am posting it here also to increase the potential number of people who will see it. I realize that this question might not be entirely in the spirit of ...
-2
votes
0answers
56 views

Jacobson radical of an indecomposable commutative ring [migrated]

Let $R$ be a commutative indecomposable ring with identity which has infinit many maximal ideals. Can we deduce that $Jacobson$ radical, $J(R)$, (the intersection of all maximal ideals)of $R$ is th ...
2
votes
0answers
141 views

Help for reference of moduli stack of fake elliptic curve

I see everywhere says the following: Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, $\mathcal{O}_B$ be a maximal order, $N$ be an positive integer coprime to $D$. ...
0
votes
1answer
187 views

Torsion in the (co-)homology of a smooth projective variety - what is known in general?

There are lots of ways in which the complex singular (co-)homology of a smooth projective variety over $\mathbb{C}$ is "special" among complex manifolds - the hard Lefschetz theorem, the Hodge ...
3
votes
2answers
483 views

Realizing a Jacobian as a projective variety

This is a follow-up to this question (and my answer thereto). Given an algebraic curve, is there a way to realize its Jacobian explicitly as a zero-set of a bunch of polynomials in $\mathbb{P}^n.$ I ...
6
votes
2answers
328 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
-1
votes
0answers
98 views

nonsingular schemes and regularity of stalks

Let $V$ be an affine variety and $P$ be a point of $V$. Zariski has proved that $P$ is a nonsingular point for $V$ if and only if the local ring $\mathcal{O}_{P}$ is a regular ring. Notice that a ...
3
votes
1answer
135 views

Can we define a height function for a variety over a finite field?

That is, is there a way to measure the complexity of a point over a finite field the same way we do it over number fields?
1
vote
0answers
96 views

In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$. Let $0$ be one of the origins of ...
0
votes
0answers
156 views

Bigness of a symplectic form on pair $(X,D)$

Let $(M,\omega_M)$ be a compact Kähler manifold. We say that a semi-positive $(1,1)$ form $\omega$ is big iff $$\int_M\omega^n>0$$. Now let we have the pair $(X,D)$ where $D$ is a divisor on ...
2
votes
0answers
97 views

Do we know when $R^if_*\omega_{Y}$ is k-th syzygy sheaf?

Let $f:Y\rightarrow X$ proper surjective morphism between smooth projective varieties. If $f$ is smooth outside a simple normal crossing divisor, we know that $R^if_*\omega_Y$ is locally free.[Kollár, ...
0
votes
1answer
84 views

Projective normality of cones over projectively normal varieties

Let $X\subseteq\mathbb{P}^n$ be a smooth subvariety, with homogeneus ideal $I\subseteq k[x_0,\ldots,x_n]$. Let $C(X)\subseteq\mathbb{P}^{n+1}$ be the projective cone over $X$, so that $C(X)$ is ...
0
votes
0answers
49 views

The trace ideal of a non zero $R$-module [migrated]

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
0
votes
1answer
188 views

When is the Thom class the Poincare dual of the zero section?

As the title suggests, when is the Thom class the Poincare dual of the zero section? For starters, it's true for the normal bundle of an immersion...
2
votes
2answers
337 views

Reference for higher categorical analogue of algebraic cycle? [closed]

Are there higher categorical analogues of algebraic cycles? What are some references? This question arise in an attempt to generalize algebraic cycles towards higher dimensional algebra. Has there ...
15
votes
2answers
685 views

Varieties with an ample vector bundle mapping to their tangent bundle

A well-known result of Andreatta and Wisniewski says: Let $X$ be a projective complex manifold whose tangent bundle $T_X$ contains an ample sub-bundle $\mathscr{E}$. Then $X$ is isomorphic to ...
6
votes
1answer
170 views

GIT quotients and automorphisms

Let $X$ be a smooth projective variety. Then we have an exact sequence: $$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$ where $Aut^{o}(X)$ and $H$ are respectively the connected ...
7
votes
1answer
336 views

Example of Genus 7 Curve whose Conormal Sheaf isn't Locally Free

Let $C \rightarrow \mathbb P^6$ be a genus 7 canonically embedded (singular) Gorenstein generically reduced curve. Are there any examples of such a curve so that the conormal sheaf $N^\vee_{C/\mathbb ...
6
votes
1answer
129 views

Varieties parametrizing skew-symmetric matrices

Let $V$ be a vector space of dimension $n$ and let us consider the projective space $\mathbb{P}(\bigwedge^2V)$ parametrizing skew-symmetric matrices. Let $M\in\mathbb{P}(\bigwedge^2V)$, for any ...
2
votes
0answers
94 views

What sorts of weights for perverse sheaves were or can be computed?

I am studying certain weights for (triangulated categories of relative) motives. Those are interesting; yet one can hardly say that they are very much explicit or effectively computable. So, I would ...
1
vote
1answer
165 views

Algebraic Geometry - Blowup

I am starting to study for my Msc dissertation and i want / have to study the Blowing up transformation in algebraic geometry. I know little about algebraic geometry but i'm a stubborn learner and so ...
5
votes
1answer
468 views

reference for “Topological algebra of Grothendieck”

I would like to have some references for Grothendieck's theory of "Topological algebra": a synthesis of homotopical and homological algebra, with special emphasis on topoi.
3
votes
1answer
166 views

Any three plane conics has a line meeting them in a configuration of points with an order 5 symmetry

Edit: In light of Jason's answer, the below asserted statement must be wrong, which means that the intersection theory argument alluded to in the background section must be wrong. What follows is the ...
0
votes
0answers
24 views

Under what conditions is the unison of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under what conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$, $$f(v) = A[v]_+ + B[-v]_+$$ surjective? Here $[.]_+$ is an elementwise ...
8
votes
2answers
275 views

Is a morphism whose all fibers are $\mathbf{P}^n$ a projective bundle?

Suppose $X\to Y$ is a morphism between varieties with all fibers isomorphic to $\mathbf{P}^n$, is $X$ a projective bundle over $Y$, i.e. $X=\mathbf{P}(E)$ for some vector bundle over $Y$?
2
votes
0answers
88 views

Cycle map and flat cycle

Let $\mathcal X\rightarrow C$ be a smooth projective morphism over an open subset of $\mathbb A_k^1$ ($k$ algebraically closed of characteristic $p>0$, one can suppose $C$ to be the spectrum of a ...
2
votes
1answer
101 views

Are cones over Grassmannianns of lines local complete intersections?

Let $X_d^N\subset\mathbb{P}^N$ be a cone over the Grassmannian of lines $\mathbb{G}(1,d)\subset\mathbb{P}^{d(d+1)/2-1}\subset\mathbb{P}^N$ with vertex a linear space $L\subset\mathbb{P}^N$ of ...
6
votes
1answer
175 views

Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$. If $\xi$ has Euler characteristic 0, then apparently there is ...
7
votes
3answers
573 views

Ranks of elliptic curves depend only on the field?

Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
4
votes
1answer
136 views

locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective if and only if every injective $R$-module is a direct sum of indecomposable injective ...
3
votes
1answer
159 views

variation on an exact sequence of logarithmic differentials

Let $X$ be a smooth projective complex variety and $D$ a divisor with simple normal crossings on $X$, with irreducible components $D_i$. If $D_1$ is one of these (smooth) irreducible components, then ...
2
votes
0answers
124 views

Intuition behind if neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$?

Is there any intuition behind the following fact? If neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$. Here, $K$ is the canonical divisor. It means the degrees ...
5
votes
2answers
344 views

What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?

Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$: \begin{eqnarray*} h: (x, y, z) &\mapsto& (x, y, xy - z) \\ u: (x, y, z) ...
4
votes
1answer
122 views

Solving over-determined system of polynomials

I am trying to solve the following over determined system of polynomials \begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, ...
8
votes
0answers
134 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

Reposted from math.stackexchange here. The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq ...
0
votes
0answers
56 views

Is there a general theory for the structure of the (semi)group generated by morphisms of an affine space $F_p^3$?

Consider an affine space $\mathbb{F}_p^3$, and assume we have a handful of morphisms $f_i : \mathbb{F}_p^3 \rightarrow \mathbb{F}_p^3$ given by $$f_i(x, y, z) =(P_i(x, y, z), Q_i(x, y, z), R_i(x, y, ...
1
vote
1answer
164 views

Solutions to system of polynomial equations over finite fields

If $P_1$, $P_2$, ..., $P_m$ are $n$-variate homogeneous polynomials of degree d over a finite field $F_q$, where $q$ is much larger than $d$, but much smaller than $n$, then do we know good lower ...
1
vote
0answers
147 views

An elliptic curve trivial over any extension unramified outside 7 and infinity?

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(K)$ is trivial for every finite extension $K/\mathbb{Q}$ with discriminant a power of $7$ ?
1
vote
0answers
60 views

Degrees of polynomials defining a Jacobian of maximal rank on a variety

Let $f_1,\ldots,f_{n-k} \in \mathbb{R}[x_1,\ldots,x_n]$ be polynomials of degree at most $d$ defining an algebraic set $A \subseteq \mathbb{C}^n$ which contains an irreducible component $V \subseteq ...
2
votes
0answers
62 views

Connected components of a certain real homogeneous space

Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ ...
0
votes
0answers
74 views

Complement and fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
2
votes
2answers
221 views

Three and a half basic questions on the Weil restriction of scalars

(This is reposted from mathstackexchange, where it received no answer so far.) I am currently trying to get familiar with the Weil Restriction functor. For a finite field extension $L|K$ it ...