## ag.algebraic geometry – Earliest reference for neighborhoods of the diagonal

This MO question asks about the algebraic cotangent space. The paper first neighborhood of the diagonal and geometric distributions by Kock claims Grothendieck initiated the study of neighborhoods of the diagonal, later adopted by Malgrange and Kumpera & Spencer.

What is the earliest explicit reference of the neighborhoods of the diagonal?

## ag.algebraic geometry – Associated point of Coherent Sheaf

That’s a question about a proof I found in E. Sernesi’s
Deformations of Algebraic Schemes on page 188:

The sheaf $$F$$ is assumes to be coherent on $$X= mathbb{P}^r$$. The question is
why the hyperplace $$H subset mathbb{P}^r$$ is assumed not to contain
any point of $$Ass(F)$$, the associated points
(https://stacks.math.columbia.edu/tag/02OI)

then the exact sequence $$0 to O(-H) to O_{P^r} to O_H to 0$$ stays
exact after beeing tensored by $$- otimes F(k)$$?

Obviously the problem is of local nature, so we can study what happens at affine
piece of $$mathbb{P}^r$$. Let $$R$$ be a ring, $$M$$ a coherent $$R$$-module
and $$f in R$$ a nonzero element which defines the hyperplace. The
assumption that $$H$$ not contains point of $$Ass(M)$$, translates to the
ring theoretic assumpion that for every associated point of $$M$$, that
is a prime $$p subset R$$ which annihalates some nonzero $$m in M$$, ie
$$p= Ann(m)$$, the element $$f$$ is not contaned in $$p$$.

Therefore the claim is that the sequence
$$0 to R xrightarrow{cdot f} R to R/(f)$$ stays exact after tensor
by $$M$$. Why that’s true?

## ag.algebraic geometry – A new simple formula is needed

The following question is related to the families of high rank elliptic curves with torsion subgroup $$mathbb{Z}/6mathbb{Z}$$.

The SageMath/Python code below produces a list of small fractions $$a$$ for which $$e=sqrt{a(a+1)}$$ is a multiple of $$sqrt{r}=sqrt{2}$$.

from time import time
t0 = time()
r = 2
listA = ()
top = 100
for n in range(-top, top + 1):
for d in range(1, top + 1):
if gcd(n,d) == 1:
a = QQ(n) / QQ(d)
e = sqrt(a * (a + 1) / r)
if (e in QQ) and a != -1 and a != 0:
listA.append(a)
print(listA)
print(time() - t0)
(-98/17, -98/73, -98/89, -98/97, -81/31, -81/49, -81/73, -81/79, -72/23, -72/47, -72/71, -50, -50/41, -50/49, -49/17, -49/31, -49/41, -49/47, -32/7, -32/23, -32/31, -25/7, -25/17, -25/23, -18/17, -9, -9/7, -8/7, -2, 1, 1/7, 1/17, 1/31, 1/49, 1/71, 1/97, 2/7, 2/23, 2/47, 2/79, 8, 8/17, 8/41, 8/73, 9/23, 9/41, 9/89, 18/7, 18/31, 25/7, 25/47, 25/73, 32/17, 32/49, 32/89, 49, 49/23, 49/79, 50/31, 50/71, 72/49, 72/97, 81/17, 81/47, 98/23, 98/71)
1.3610780239105225

Given some $$a$$, I would like to derive a simple formula to produce a different value $$b$$ in the same list. I would measure the complexity of the formula by the sum $$s$$ of degrees of its numerator and denominator.

By analyzing the high rank $$mathbb{Z}/6mathbb{Z}$$ families, three simple formulas were derived to date. Note that these formulas are true for any value of $$r$$, not just $$r=2$$.

$$(1)$$ $$b=-frac{(a+1)}{(1-3a)}$$; $$s=2$$

$$(2)$$ $$b=frac{a(a+1)}{(1-3a)}$$; $$s=3$$

$$(3)$$ $$b=-frac{(5a+1)^2}{(3a-1)^2}$$; $$s=4$$

Applying $$(2)$$ and $$(2)$$ consecutively:

$$(4)$$ $$b=frac{a(a+1)(a^2-2a+1)}{(-1+6a+3a^2)(-1+3a)}$$; $$s=7$$

For a higher value of $$s$$:

$$(5)$$ $$b=-frac{(1+2a+5a^2)^2}{(-1+6a+3a^2)^2}$$; $$s=8$$

My questions:

1. Is it possible to derive some new formula(s) for $$sle6$$?
2. Is there a way to write a smarter/faster code, e.g., by skipping some values of $$n$$ and/or $$d$$ for a given value of $$r$$?

## ag.algebraic geometry – Does self-product of universal hypersurfaces have dlt singularites?

Let $$ngeq2,dgeq 2n+1$$ be integers, let $$mathcal{X}_{n,d}to|mathcal{O}_{mathbb{P}^n}(d)|$$ be the universal family of hypersurfaces. The total space $$mathcal{X}_{n,d}$$ is smooth as it is a projective bundle over $$mathbb{P}^n$$.

The self-product $$T:=mathcal{X}_{n,d}times_{|mathcal{O}_{mathbb{P}^n}(d)|}mathcal{X}_{n,d}$$ is a complete intersection in $$|mathcal{O}_{mathbb{P}^n}(d)|_{}times mathbb{P}^n_{(X_0,cdots,X_n)}timesmathbb{P}^n_{(Y_0,cdots,Y_n)}$$, defined by $$sum a_IX^I=0,sum a_IY^I=0$$. It has singularities in codim at least $$4$$, therefore it is $$mathbb{Q}$$-factorial.

Do we know if $$T$$ have divisorial log terminal singularities? (for complete intersections, dlt equivalent to canonical singularities, rational singularities) If not, can we say anything about them?

## ag.algebraic geometry – How to show analytification functor commutes with forgetful functor?

Also in ME.

Let $$k$$ be a field complete with respect to a non-trivial non-archimedean
absolute value (so that rigid $$k$$-space makes sense). Denote $$K$$ a finite field extension of $$k$$.

Denote $$Xrightsquigarrow X^{mathrm{an}/k}$$ the analytification functor from the category of locally of finite type $$k$$-schemes to the category of rigid $$k$$-spaces.
Similarly there is an analytification functor $$Xrightsquigarrow X^{mathrm{an}/K}$$ over $$K$$.

There is a well-defind forgetful functor $$S:Xrightsquigarrow X$$ from $$K$$-schemes to $$k$$-schemes ($$S$$ represents schemes) and a forgetful functor $$R:Yrightsquigarrow Y$$ from rigid $$K$$-spaces to rigid $$k$$-spaces ($$R$$ represents rigid).

Let $$X$$ be a locally of finite type $$K$$-scheme. I believe that $$S(X)^{mathrm{an}/k}cong R(X^{mathrm{an}/K})$$ as rigid $$k$$-spaces. The universal property induces a canonical map $$R(X^{mathrm{an}/K})to S(X)^{mathrm{an}/k}$$ but I cannot show it is an isomorphism. A proof or reference would be nice.

p.s. the idea comes from proving absolute/relative Frobenius morphism commutes with analytification, but I first need to make sure the maps have the same source.

## ag.algebraic geometry – K-equivalence => isomorphism of Chow motives?

An old conjecture of Bondal-Orlov-Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata’s paper for definitions. In particular this applies to birational Calabi-Yau varieties. See this MO question for the Bondal-Orlov formulation of the conjecture and the known cases. These conjectures are mostly open, and considered to be an important bridge between derived categories and birational geometry.

Derived categories and Chow motives play the role of universal cohomology theories, in noncommutative, and commutative worlds respectively. Do we expect that K-equivalence implies isomorphism of rational, or even integral Chow motives?

Example. Integral Chow motives of varieties related by a standard flop are isomorphic: paper by Q. Jiang. This was the motivation for the question.

Remarks. In many cases, D-equivalence is known to imply isomorphisms for rational Chow motives, but not for integral ones: examples can be found among K3 surfaces. It is also known that D-equivalence does not imply K-equivalence, even for rational surfaces.

## ag.algebraic geometry – The image of a curve under the multiplication endomorphism of its Jacobian

Let $$X$$ be a complex smooth projective curve of genus $$ggeq 2$$. Embed $$X$$ in its Jacobian
$${rm{J}}(X)cong{rm{Div_0}}(X)/{rm{Div_p}}(X)$$ where $${rm{Div_0}}(X)$$ is the group of degree zero divisors and $${rm{Div_p}}(X)$$ is the subgroup formed by principal divisors. The embedding can be in the form of for instance
$$pmapsto text{the divisor }p-p_0text{ modulo }{rm{Div_p}}(X)$$
where $$p_0$$ is an arbitrary base point. What kind of curve the image of $$Xsubset {rm{J}}(X)$$ under the multiplication by $$n$$ map ($$n>1$$ an integer) $$(n):{rm{J}}(X)rightarrow{rm{J}}(X)$$ is? Since the endomorphism is locally biholomorphic, I guess the image is smooth if $$(n)$$ restricts to an injective map on $$X$$. In general, what kind of singular curves may arise in this way? In what situations $$(n)$$ sends $$X$$ to another smooth curve inside $${rm{J}}(X)$$?

Here is a special case that I can (almost) analyze: Suppose $$nleq g$$. Then distinct points $$p,p’in X$$ have the same image under $$(n)$$ iff $$np-np’$$ is a principal divisor and this happens only when $$p$$ and $$p’$$ are Weierstrass points. Thus the restriction of $$(n)$$ is generically one-to-one and so the image is singular if there exists a principal divisor of the form $$np-np’$$ (e.g. when $$X$$ is of the form $$y^n=f(x)$$ with $$f$$ square-free).

## ag.algebraic geometry – the map on divisor class groups induced by restriction to a toric subvariety

Let $$X$$ be a (say, complex) toric variety acted upon by a torus $$T$$ and defined by a fan $$Sigma$$ in the cocharacter lattice $$N=mathrm{Hom}(mathbb{C}^times, T)$$, and let $$M$$ be the character lattice. For any cone $$sigma in Sigma$$ put $$M(sigma) = sigma^perp cap M$$, $$N(sigma) = mathrm{Hom}(M(sigma), mathbb{C}^times)$$. There is a natural projection $$N to N(sigma)$$. Then the closure of the orbit corresponding to $$sigma$$ has the structure of a toric variety with respect to the quotient torus with the cocharacter lattice $$N/N(sigma)$$ and given by the fan $$Star(sigma)$$ consisting of the images in $$N(sigma)$$ of the cones of $$Sigma$$ containing $$sigma$$. Note that the closed embedding $$X_{Star(sigma)} to X$$ is generally not a toric morphism, since the dense toric orbit of $$X_{Star(sigma)}$$ does not intersect the dense toric orbit of $$X$$.

My question is: is there a way to describe the restriction map $$mathrm{Cl}(X) to mathrm{Cl}(X_{Star(sigma)})$$ in terms of the fans $$Sigma$$ and $$Star(sigma)$$?

## ag.algebraic geometry – On the Dimension of the Dual Variety of a Singular Hypersurface

I was primarily interested in the following question. Let $$ngeq 3$$, and let $$Xsubset mathbb{P}^n$$ be a degree $$d$$ hypersurface. Assume that its singularity locus $$S$$ (with reduced structure) is irreducible and smooth of dimension $$k$$. Is it true that
$$dim S^vee >dim X^vee ?$$
The statement is true for quadratic hypersurfaces and hypersurfaces with isolated singularities. I am wondering if in general this holds.

I was reading the book Discriminants, Resultants, and Multidimensional Determinants by Gelfand-Kapranov-Zelevinksy. In the first chapter they introduce the Katz dimension theorem, which computes the dimension of the dual varieties via local coordinates. But I found it difficult to use, since it’s quite hard to write down the local coordinates. Is there any comments on the Katz dimension theorem, especially on how to use it ?

## ag.algebraic geometry – Picard group of \$(SL(n)times SL(m))\$-orbits

Let $$mathbb{P}^N$$ be the projective space of $$ntimes m$$ matrices with complex entries modulo scalar. Consider the $$(SL(n)times SL(m))$$-action on $$mathbb{P}^N$$ given by $$((A,B),Z)mapsto AZB^{T}$$. Now, consider the matrix
$$J_k = left( begin{array}{cc} I_{k} & 0 \ 0 & 0 end{array} right)$$
and let $$X_ksubsetmathbb{P}^N$$ be the orbit of $$J_k$$. I would like to ask if anyone knows how to compute the Picard group of $$X_k$$.

Thank you very much.