## New answers tagged ag.algebraic-geometry

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### Motivation behind spectral sequences

I suspect a previous comment of mine led to this question, so let me say a few words here. The basic problem is this: Suppose $(A^\bullet, F)$ is a filtered complex, then one wants to relate the (co)...

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### Are algebraic groups over algebraically closed fields Cohen-Macaulay?

In arbitrary characteristic they are locally complete intersection, hence Cohen-Macaulay: this is in SGA3, Exposé VII$_B$, Cor. 5.5.1.

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### Are algebraic groups over algebraically closed fields Cohen-Macaulay?

If all you want is Cohen-Macaulay, this is straightforward, and your sketch proves the result. Every locally finite type scheme over a field has a maximal open subscheme that is Cohen-Macaulay (...

Community wiki

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### Two morphisms possess the same Viehweg's variation

Since the definition only depends on the general fiber, and $\beta$ is birational, one may assume that $\beta$ is actually and isomorphism.
So, then $L$ is defined as a subfield with minimal ...

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### Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$

According to @Donu Arapura's comment, I give an answer of my understanding, whether it is correct or not, please comment below.
By the definition of Frölicher spectral sequence degenerating at $E_1$, ...

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### Is the Segre embedding of two real varieties a real variety?

$\mathrm{Seg}(X\times Y)$ is a real projective variety since the full Segre map is an isomorphism of real algebraic varieties onto its image.
As for your second question, I think the answer is "...

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### How does the rational curve behave under the group action?

The double covering $\pi_2 \colon X \to \mathbb{P}^2$ is the anticanonical morphism, therefore the image of $l_1 = \pi_1^{-1}(l)$ under $\pi_2$ is a cubic curve, hence $\pi_2^{-1}(\pi_2(\pi_1^{-1}(l)))...

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### Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?

No.
Working on projective space, consider a composition $$ \mathcal O \to \mathcal O \oplus \mathcal O/\mathcal O(-1) \to \mathcal O $$ where $\mathcal O/\mathcal O(-1)$ is the constant sheaf on a ...

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6
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### Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

If you assume that $\mathcal{P}$ and $\mathcal{P'}$ are semiorthogonal, this is true. The easiest way to see this is by looking at the singularity category. If $X$ has one node, (the idempotent ...

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### Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$

For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex ...

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### Rational curves on the image of the pluricanonical maps

Not only could $Y_m$ contain a rational curve for all $m$, $Y_m$ could be a rational curve for all $m$.
Take $C$ a hyperelliptic curve, $E$ an elliptic curve, $\tau$ the hyperelliptic involution on $C$...

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### Topologies in the vicinity of Euclidean space

Under reasonable assumptions about $\Sigma$ the answer is yes. For example if $\Sigma$ is smooth and compact $(n-m)$-dimensional submanifold of $\mathbb{R}^n$ and it has trivial normal bundle*, that ...

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### Variant of Wahba's problem

This is not a rigorous proof, just some geometrical considerations. You can rephrase your problem to a problem on the unit sphere. The normalized $v_k$ build a triangle on the unit sphere. With the ...

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### What exactly do the standard conjectures in characteristic zero refer to?

To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $...

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### What exactly do the standard conjectures in characteristic zero refer to?

Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption ...

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### D-modules on singular varieties; forgetful functors, and t-structures

Just in case anyone else has the same question in the future, here is an answer to both questions as long as we're willing to replace quasi-coherent sheaves with ind-coherent sheaves. If we restrict ...

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### Realize as homology a given polynomial ring

This seems related to the Steenrod problem (see below). As far as I know it's still open. Here are a couple of references; maybe the Anderson-Grodal one is closer to what you're looking for. N.B. I ...

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### About Fulton's Intersection theory Appendix Lemma A 4.1

No. Whereas $A/I\otimes_A B=B/IB$ by right-exactness $I\otimes_A B\to IB$ is not an isomorphism in general. The exact sequence
$$0\to I_{i-1}\to I_i\to I_i/I_{i-1} \to 0$$
only gives rise to the right ...

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### Points on curves of genus 3

No. Note that $P\neq Q$ since $i$ is fixed-point free. Since $i^*K=K$, one would have $5P+3Q\equiv 3P+5Q$ (where $\equiv$ means linear equivalence), hence $2P\equiv 2Q$, so $P$ and $Q$ are Weierstrass ...

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### Artin vanishing for D-modules (i.e., when is $f_+$ t-exact?)

The functor $f_+$ by definition is the composite of a right t-exact functor (tensor product with the transfer bimodule) and a left t-exact functor (the derived pushforward of sheaves). In the case $f$ ...

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### Trigonometric Diophantine equation

This is really a question about solving multivariable polynomials in roots of unity, which in turn can be deduced from finding all linear combinations of roots of unity with certain coefficients that ...

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### Minor Lefschetz principle

The correct statement is that the following are equivalent, for a sentence $\varphi$ in the first-order language of fields.
$\varphi$ is true in some algebraically closed field of characteristic $0$.
...

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### Definition of discrepancy

If the morphism is not proper then an exceptional divisor can be linearly equivalent to zero. Indeed locally near a point of any exceptional divisor this must be true. Then localising gives such a ...

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### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Section 3 of the following paper of Beauville should answer your question: https://arxiv.org/abs/alg-geom/9701019
In particular, it is shown there that, up to replacing the compactified Jacobian by a ...

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### Library/Database of parametric polynomial systems

There is a database of polynomial systems, which comes with PHCpack by Jan Verschelde.

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### Why are local systems and representations of the fundamental group equivalent

Here is the way I think of the correspondence between locally constant sheaves and representations of the fundamental group, and how I like to tell my students about it when I introduce it in class (...

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### How to simplify this homotopy totalization coming from an arc-cover into a pullback?

Let $X$, $Y$, and $Z$ be Kan complexes. We wish to show that
$X\times_YZ$ in $\mathrm{Spc}$ can be computed as the limit of the diagram
$$(*)\qquad X\times Z\rightrightarrows X\times Y\times Y\times Z\...

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3
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### Blowing up of a singular subvariety

Possibly the simplest example is to consider the blowup of a reduced and irreducible curve $C$ in a smooth 3-fold $X$ with a point $P\in C\subset X$ which is locally analytically isomorphic to
$$ 0\in ...

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2
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### Intersection pairing and birational morphisms

First note $\overline{f(C \cap V)} = f(C)$, since $f$ is closed and $C$ (and hence also $f(C)$) is irreducible. Also $f$ induces a birational map $C \to f(C)$, so $f_* [C] = [f(C)]$ where $[\cdot]$ ...

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### Fibers of the coarse moduli space map

You're guess is correct. Here's an approach: For any stack and any global section of the structure sheaf, we can define the vanishing set in the obvious way: It's $R$-points are the $R$-points of the ...

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### Hermitian holomorphic line bundle and curvature Chern form in Demailly's book

Thanks to the aid of @Gunnar Þór Magnússon, I will write down my understanding of Demailly's proof, if there is anything unclear, please comment below.
Let $\cup_{i\in I}U_i$ be a covering of $X$ such ...

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5
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Accepted

### Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

The answer to your question at the end is negative. In fact, $h^2(D, j_*F)= h^2(D, j_* F)=0$. In fact, the cohomology of a sufficiently small disc around a point in any complex variety, with ...

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### To whom is Bézout's theorem for varieties due?

I'll expand on the comment by @red_trumpet. Quoting from p. 152 of Fulton's book: "The result of Example 8.4.6 was discovered and proved with MacPherson and Lazarsfeld, in answer to a question of ...

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### Gorenstein varieties: why the two definitions are equivalent?

As Donu mentioned, Gorenstein can be defined as Cohen-Macaulay and such that the canonical=dualizing sheaf is a line bundle. The point is that the dualizing complex is quasi-isomorphic to a sheaf if ...

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### Given an integer $N$, find solutions to $X^3 + Y^3 + Z^3 - 3XYZ \equiv 1 \pmod{N}$

Here is a method to find your triples that is not rapid, perhaps trivial, and highly conjectural. Hopefully it is still of some interest. First, find positive integers $x$, $y$, and $z$ that solve $...

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### Global sections of a line bundle on a reducible complex space

In any case $H^0(S,L)$ is a subspace in $\oplus H^0(V_i,L)$, but the way it sits there depends on the way the components $V_i$ are glued together. In the simplest case where they for a simple normal ...

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### Bounded generation of group by unipotent radicals of opposite parabolic subgroups

I hope you will permit me to write $P^+$ in place of $P$. Put $U^\pm = R_u(P^\pm)$.
Yes, at least in the split case.
Suppose first that $P^+$ and $P^-$ are minimal. Put $T = P \cap P^-$. By working ...

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### Blowing up of a singular subvariety

Just blow up the singular
point in a variety $X_1$ which is obtained from
a smooth, irreducible manifold $X$ by identifying
points $x$ and $y$. The blow-up divisor
is ${\Bbb P} T_xX\coprod {\Bbb P}...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

Here is an additional comment. Every homogeneous polynomial can be regarded as a symmetric tensor. Quadrics correspond to symmetric matrices; cubics correspond to order $3$ tensors; in general, a ...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

Also over the rationals (as @KConrad might have mentioned!), the theory of quadratic forms is nicer than the theory of higher forms:
Hasse’s principle holds for quadratic forms: if a quadratic form ...

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### Absolute integral closure of local UFD

For the finite extensions as you describe. There is no chance they will always be flat even when $R$ is a regular ring of characteristic $p > 0$. Indeed, there are plenty of finite extensions $R \...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

This is really more of a remark. The zero set of a homogeneous polynomial $f(x_0,\ldots, x_n)$ over a field, is a hypersurface in the projective space $\mathbb{P}^n_k$. These objects have been ...

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### Complete surfaces in $M_g$

If $d:Y\to X$ is an étale double cover, then each point $p$ of $X$ gives a pair $\{a_p,b_p\}=d^{-1}(p)$ of distinct points in $Y$.
Given such a $\{a_p,b_p\}$ there are finitely many double covers $e:...

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### Complete surfaces in $M_g$

Theorem 2.33 of Moduli of curves by Harris and Morrison gives a construction with weaker bounds.
The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 ...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

The study of forms appears to grow "exponentially harder" with their degree as the following example seems to indicate.
Let us work over the field of complex numbers since the following ...

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Accepted

### Chern class of torsion sheaf support on a point

If you can figure out the Chern character of $i_*(\mathcal{O}_Z)$ where $Z = \text{Spec}(A/m)\hookrightarrow X$ by GRR then you use the following short exact sequence:
$$0 \rightarrow m/m^2 \...

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Accepted

### The kernel of $H^{\bullet,\bullet}_{\bar\partial}(X)\to H_A^{\bullet,\bullet}(X)$

Let $\alpha$ be a $\bar{\partial}$-closed form. Denote its Dolbeault cohomology class by $[\alpha]_{\bar{\partial}}$ and its Aeppli cohomology class by $[\alpha]_A$; note that the map $g$ is given by $...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

Cubic forms are much more complicated than quadratic forms, so it may not be possible to develop a theory to end it all.
One direction of cubic forms is cubic composition laws, similar to Gauss's ...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

This is perhaps more of a comment than a full answer, but even ternary cubic forms are not well understood, and results about them are at the cutting edge of current research. See for example Ternary ...

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### There is a nice theory of quadratic forms. How about cubic forms, quartic forms, quintic forms, ...?

This is an extended comment on KConrad's discussion of symmetry groups. We can think of $k$-forms on a vector space $V$ (homogeneous polynomials of degree $k$) abstractly as elements of the symmetric ...

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