All Questions
Tagged with duality ag.algebraic-geometry
13 questions with no upvoted or accepted answers
14
votes
0
answers
899
views
Frobenius upper shriek/flat of a dualizing complex
Let $X$ be a separated connected scheme of characteristic $p > 0$. I am going to assume that $F : X \to X$ (the absolute Frobenius) is a finite map. This condition is called being $F$-finite.
...
6
votes
0
answers
163
views
Explicit computations of Serre duality for elliptic curves
I have an elliptic curve $E$ defined over a ring $R$, I want to compute the pairing
$$
H^1(E,\mathcal{O}_E)\times H^0(E, \Omega_E^1){\rightarrow}R.
$$
Clearly we have that $H^0(E, \Omega_E^1)=R \...
- 109
6
votes
0
answers
903
views
Is there a direct way to compute the higher derived image sheaves of a family of $\mathbb{P}^n$s?
Let $V\rightarrow Y$ be a vector bundle of rank $n+1$ over $Y$, with $Y$ reasonably nice (I care about the case of smooth, irreducible affine). Let $X=\mathbb{P}(V)$ be the projectivization of $V$, so ...
- 13k
5
votes
0
answers
383
views
Serre functors for non-proper categories
One usually defines a Serre functor to be a functor on a $k$-linear category $\mathcal{C}$ which has finite dimensional $Hom$s over $k$. In that case, the standard definition is that a Serre functor $...
- 512
5
votes
0
answers
141
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 \...
- 1,593
5
votes
0
answers
449
views
Dual of a weighted projective space
I have a fairly good understanding of what the dual of a projective space is. I am currently interested in weighted projective space but I haven't found anything on the construction of its dual space ...
- 131
4
votes
0
answers
166
views
Is the Serre dualizing complex local in the analytic topology?
There is a Serre dualizing complex $S_X\in D^b Coh(X)$ for any scheme $X$ of finite type. For proper schemes, it is characterized as the sheaf defining the right adjoint to derived global sections, ...
- 7,972
4
votes
0
answers
347
views
How is the restriction of the dualizing sheaf to an irreducible component related to the dualizing sheaf of the component?
$\DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\hom}{\mathcal{Hom}} \DeclareMathOperator{\ox}{\mathcal{O}_X}$Let $f:X \to Y$ be a proper morphism. In section 6.4. of Liu's book he introduces ...
- 435
4
votes
0
answers
264
views
Does GKZ's reflexivity theorem imply the Plucker formula?
Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
- 3,187
2
votes
0
answers
167
views
Open nature of $\mathcal{H}om$ functor/upper semi-continuity of $\operatorname{Ext}^i$
Let $k$ be an algebraically closed field, $T$ a $k$-scheme (can assume connected) and $X$ a projective variety over $k$. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k T$ flat over $T$. ...
- 1,981
1
vote
0
answers
127
views
Local freeness of dualizing sheaf
I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...
- 613
1
vote
0
answers
195
views
Gross-Hopkins duality
$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
- 448
1
vote
0
answers
245
views
Dual varieties and nodal sections
Let $X$ be a(n even dimensional) smooth complex projective variety in $\mathbb{P}^N$, and let $X^{\vee}$ be its dual variety; up to an higher degree Veronese embedding of $X$, I assume that $X^{\vee}$ ...
- 828