## 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.

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 étale 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 ?

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