# Tag Info

1 vote

### 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 (...
1 vote

### 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 ...
• 41.1k

### 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 $E_1$, ...
• 167

### 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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• 12.2k

### 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 ...
• 15.8k

### 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 ...
• 2,680

### 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 ...
• 17.8k

### 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 ...
• 34.8k
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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 $f$ ...
• 6,439

### 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 ...

### 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 $0$. ...
• 3,607

### 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 ...

### 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 ...
• 1,232
1 vote

### Library/Database of parametric polynomial systems

There is a database of polynomial systems, which comes with PHCpack by Jan Verschelde.
• 26.8k

### 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 (...
Accepted

• 791
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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 ...
• 119k
1 vote

### 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 ...
• 167
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### 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 ...
• 119k

### 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 ...
• 318
1 vote

### 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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### 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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1 vote

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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 ...

### 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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### 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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