ag.algebraic geometry – Vanishing of intermediate cohomology for a multiple of a divisor

Let $S subset mathbb P^3$ be a smooth projective surface (over complex numbers). Let $C$ be a smooth hyperplane section. Let $Delta$ be a non-zero effective divisor on $S$ such that $h^1(mathcal O_S(nC+Delta))=0, h^1(mathcal O_S(nC-Delta))=0$ for all $n in mathbb Z$. Then my question is the following :

In this situation can we say that: $h^1(mathcal O_S(m Delta))=0$ for $m geq 2$? Can we impose any condition so that this happens?

Any help from anyone is welcome.

ag.algebraic geometry – The classifying stack of $PGL(2)$ and the moduli space of genus zero curves

The classifying stack of $PGL(2)$ is the stack quotient $(operatorname{Spec} k/PGL(2))$ where $PGL(2)$ acts trivially on $operatorname{Spec} k$.

Since $(operatorname{Spec} k/PGL(2))$ is a quotient stack, it is an algebraic stack with smooth covering map $q: operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ defined by the pair $(PGL(2), rho)$ where $rho: PGL(2) times operatorname{Spec} k to operatorname{Spec} k $ is the action map.

On the other hand, $(operatorname{Spec} k/PGL(2))$ is also the algebraic moduli stack of genus-zero curves. Since any map $operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ is identified by Yoneda’s lemma with an object in the fiber over $(operatorname{Spec} k/PGL(2))(operatorname{Spec} k)$ a smooth covering map $q’:operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ could just as well have been defined by $mathbb{P}^1$.

How can I recover the map $q$ defined by $(PGL(2), rho)$ from the smooth covering map $q’$ defined by $mathbb{P}^1$?

I know that a map $operatorname{Spec} k to (operatorname{Spec} k/PGL(2))$ should correspond to more than just an object in
$(operatorname{Spec} k/PGL(2))(operatorname{Spec} k)$—maybe an object plus automorphism of that object.

ag.algebraic geometry – Automorphism of a stack morphism

Let $X$ be an algebraic stack and let $f: S to X$ be a smooth covering of $X$ by a scheme $S$.

Motivation: Forgetting about stacks for a moment and going back to covering spaces: Given a covering map $f: Z to Y$, with $Z$ connected and $Y$ locally connected, then $operatorname{Aut}(Z/Y)$ acts properly and discontinuously on $Z$. Moreover, if $operatorname{Aut}(Z/Y)$ acts transitively on a fiber of $p in Y$, then the covering is a $G=operatorname{Aut}(Z/Y)$-covering in the sense that $f: Z to Y cong Z/operatorname{Aut}(Z/Y)$ is a quotient map.

I am interested in making an analogous statement in the case of a smooth cover $f:S to X$ of an algebraic stack $X$. (Of course, dropping words like properly and discontinuously and keeping in mind that $f$ is not finite ‘etale and thus not a covering map in the above sense).

In particular, I want to describe $operatorname{Aut}(S/X)$. The “elements” of $operatorname{Aut}(S/X)$ are maps $phi: S to S$ such that $f circ phi =f$.
On the other hand, $f: S to X$ can be identified with a unique object $s in X(S)$ (up to $2$-isomorphism?) by the 2-Yoneda lemma and so it seems like $operatorname{Aut}(S/X)$ should have an interpretation in terms of the groupoid of maps $s to s$ lying over a given $phi: S to S$.

That is all very abstract, so let us just suppose that the elements of $X(S)$ have some geometric interpretation, for example, the object $s in X(S)$ is a family of genus g curves $C$ over a scheme $S$.

(1). Is there an interpretation of the groupoid $s to s$ lying over $phi: S to S$ in terms of a group automorphisms of $C$ over $S$?

(2). Moreover, how would this group act on a “fiber” $S times_{X,g} T$ (a sheaf on the category $Sch/S times T$?) over $g: T to X$?

ag.algebraic geometry – On Cayley Bacharach property

Let $Z$ be a zero-dimensianl locally complete intersection in $mathbb{P}^3$ such that $Z$ satisfies Cayley Bacharach property for $mathcal{O}_{mathbb{P}^3}(n)$. Then if length of $Z$ is smaller than $h^0(mathcal{O}_{mathbb{P}^3}(n))$, then $Z$ fails to impose independent conditions on sections of $mathcal{O}_{mathbb{P}^3}(n)$. Let $Z^{‘}$ be the reduced part of $Z$.

Question: Does $Z^{‘}$ also fails to impose independent conditions on sections of $mathcal{O}_{mathbb{P}^3}(n)$ ?

ag.algebraic geometry – Hodge conjecture for rationally connected/Fano hypersurfaces

We know due to work of Lewis, Murre and others that the rational Hodge conjecture holds for smooth projective hypersurfaces in $mathbb{P}^5$ of degree at most $5$. Does a similar result hold in higher dimension? In particular, is it known if the Hodge conjecture holds for smooth, projective hypersurfaces in $mathbb{P}^{2n+1}$ of degree at most $2n+1$ for some other values of $n$ greater than $2$?

ag.algebraic geometry – Kernel sheaf of evaluation map

I have some questions on the kernel sheaf of the evaluation map.

Let $F$ be a globally generated rank $2$ torsion free stable sheaf on a smooth projective variety $X$. Then the evaluation map $ev: H^0(X, F)otimes mathcal{O}_X to F$ is surjective. Denote $K:=Ker(ev)$, Then I wonder that:

(1) Is $K$ always locally free?

(2) What can we talk about the stability of $K$?

Thanks for any help.

ag.algebraic geometry – Local rings $R subseteq S$ with $m_RS=m_S$

Let $R subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $m_S$ (= the maximal ideal of $S$).

Further assume that $R$ and $S$ are $mathbb{C}$-algebras, $R subseteq S$ is flat and algebraic but not integral,
where algebraic non-integral means: Every element of $S$ satisfies a polynomial with coefficients in $R$, with non-invertible leading coefficient.

Could one find an example of such rings?

Unfortunately, the examples I find are integral, for example:
$R=mathbb{C}(x(x-1))_{(x(x-1))}$, $S=mathbb{C}(x)_{(x)}$.

Remarks:

(i) I am interested in both cases where $R$ and $S$ have the same fields of fractions or different fields of fractions.

(ii) Recall the following results, which are not applicable here, since I assume that $R subseteq S$ is non-integral:
If $A subseteq B$ is integral and flat, then $A subseteq B$ is faithfully flat, and if in addition $Q(A)=Q(B)$ (same fields of fractions), then $A=B$.

Relevant questions: a, b and c.
Also asked in MSE.

Any hints and comments are welcome; thank you.

ag.algebraic geometry – Is it true that, $ H^{2r} ( X , mathbb{Q}_{ ell } (r) ) simeq H^{2r} ( overline{X} , mathbb{Q}_{ ell } (r) )^G $?

Let $ k $ be a field and let $ X $ be smooth projective variety over $ k $ of dimension $ d $.
We denote by $ overline{X} = X times_k overline{k} $ the base change of $ X $ to the algebraic closure $ overline{k} $.
The Galois group $ G = mathrm{Gal} ( overline{k} / k ) $ then acts on $ overline{X} $ via the second factor.

Is it true that, the $ ell $ – adic ├ętale cohomology vector space, $ H^{2k} ( X , mathbb{Q}_{ ell } (r) ) $ verifies, $$ H^{2r} ( X , mathbb{Q}_{ ell } (r) ) simeq H^{2r} ( overline{X} , mathbb{Q}_{ ell } (r) )^G $$ ? How to prove that ?

Thanks in advance for your help.

ag.algebraic geometry – Tensors of minimal rank in Schur modules $S_{lambda}V subset V^{otimes |lambda|}$

It is well known that for a vector space $V$ with $dim(V)=n+1$ the $GL(V)-$module $V^{otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{lambda}V$, where $lambda=(lambda_1,dots,lambda_r)$ is a partition of $d$ (with suitable properties).
For example in the case $d=2$ we have that $$V otimes V=Sym^2(V) oplus bigwedge^2(V)$$ associated to partitions $(2,0)$ and $(1,1)$.
Let $S_{n,d}$ denote the Segre variety of tensors in $V^{otimes d}$ parametrizing tensors of rank $1$, i.e. tensors of the form $$T=v_1 otimes dots otimes v_d$$
We say that a tensor $T$ has rank $r$ if $T=alpha_1T_1+dots alpha_rT_r$ where $T_i$ are tensors of rank $1$ and $r$ is the minimum among such expressions. Similar we define the border rank $bRank(T)=r$ if $T$ can be written as a limit of tensors of rank $r$.

Now for the example where $d=2$ we have that $$Sym^2(V) cap S_{n,d}=V_{n,d}$$ is the Veronese variety parametrizing symmetric tensors that are power of linear forms, i.e. $T=v^2$. In this case the minimal rank of tensors $T in Sym^2(V)$ is exactly $1$, the best possible. However for the second module we have that $$bigwedge^2(V) cap S_{n,d}= emptyset$$
In paritcular elements $T in bigwedge^2(V)$ can be identified with skew matrices and so the minimal possible rank of a skew tensor $T in bigwedge^2(V)$ is $rank(T)=2$. In the case of matrices border rank and rank coincide and so there is no distinctions.

$textbf{My question now is the following}$: given $d$ and $lambda$ a partition with $|lambda|=d$, what can we say about the minimal rank and border rank of tensors $T in S_{lambda}V$? Is there an explicit description of the minimal ranks or even a bound as functions of $d$ and $lambda$?

I was searching for a reference about this fact/computations but I was not able to find a proper one on the internet. Maybe you can help me.
Thanks in advance.

ag.algebraic geometry – Functorial isomorphisms

We know that a priesheaf on category $ mathcal{C} $ , $ mathcal{F} $ constructed a colimit preasheaf $ mathcal{F}^{+} $ defined by $breve{C}$ech cohomology then we have the following theorem said
$ mathcal{F}^{sharp} = (mathcal{F}^{+})^{+} $ is a sheaf and the canonical map induces a functorial isomorphism $ Hom_{PSh(mathcal{C})}(mathcal{F} , mathcal{G}) $ = $ Hom_{Sh(mathcal{C})}(mathcal{F}^{sharp} , mathcal{G}) $ ,
for all $ mathcal{G} $ in $ Sh(mathcal{C}) $ .

My question is that , if we can find a functorial isomorphism between
$ Hom_{PSh(mathcal{C})}(mathcal{F}^{+} , mathcal{G}) $ and $ Hom_{Sh(mathcal{C})}(mathcal{F}^{sharp} , mathcal{G}) $ ?

And similarly question for $ Hom_{PSh(mathcal{C})}(mathcal{F} , mathcal{G}) $ with $ Hom_{Sh(mathcal{C})}((mathcal{F}^{sharp})^{+} , mathcal{G}) $ .