9

As an addendum to Donu's answer, I quote from the Table des matieres of Dix Exposés: IX GROTHENDIECK (Alexander), Crystals and the De Rham Cohomology of schemes (notes by I. Coates and O. Jussila), IHES Decembre 1966, 54 p. So it is certainly Dix Exposés. And "I. Coates" is most likely John Coates. According to Wikipedia Coates was born in '45 (so he ...


6

As Will Sawin said in his comment, this is true using Chevalley's theorem. A sketch goes as follows. Suppose that $X_\xi \to \mathrm{Spec}(\kappa(\xi))$ is smooth. Let $N \subset X$ be the nonsmooth locus. For a morphism of finite presentation, the smooth locus is retrocompact open so $N$ is constructible. Thus by Chevalley's theorem $f(N)$ is constructible ...


6

I think your hunch is correct that Google is in error, and that "Crystals and the de Rham cohomology of schemes" refers to the article in Dix Exposés. I doubt there is a book by the same name and author. There is also "On the de Rham cohomology of algebraic varieties" but that's different. The first is a long proposal for crystalline cohomology (which hadn'...


6

From my point of view, Riemann-Hilbert and non-abelian Hodge are really two independent statements - though the statement of the latter may be wound up in the former in some sense. There are three different types of objects at play: 1) Higgs bundles (Dolbeault), 2) flat connections (de Rham), 3) reps of $\pi_1$ (Betti). Riemann-Hilbert relates 2) ...


5

Tame Artin stacks (in the sense of Abramovich, Olsson and Vistoli, https://math.berkeley.edu/~molsson/tame.pdf) with quasi-projective moduli spaces will have property 2: the line bundle is the pullback of an ample line bundle on the moduli space. As to property 1, a line bundle will not be enough, but you can get a version for quotient tame Artin stacks ...


4

About question 2: I think that the software Singular has this feature; it's well-documented, and if you look for resolution graph you should find the reference. About question 1: well, I must admit that that algorithm is not pleasant, and that it took me a while to work out the case of $\Sigma(2,3,4)$. Anyway, here we go. I can't draw graphs here (but if ...


3

It is certainly true that descent implies hyperdescent whenever $\mathcal C$ is a $n$-category for some $n<\infty$ (it wasn't clear from your question whether you knew this or not). This is because, for any $\infty$-site $\mathcal A$: A presheaf $F:\mathcal A^{op}\to\mathcal C$ is a sheaf or hypersheaf if and only if $\mathrm{Map}(c, F(-)):\mathcal A^{op}...


3

The set of solutions to the $S$-unit equation for $k(X)$ is finite. Let me explain why. (You can "theoretically" find all solutions, as the finiteness eventually boils down to the "effective" finiteness result of de Franchis-Severi on maps of curves.) Let $k$ be a number field, let $X$ be a smooth projective geometrically connected curve over $k$, let $S$ ...


3

Doc, you are right but only amorally. You need to replace the tangent vectors with jets to capture the behavior of your cone. Let $I(Y)$ be the ideal of zeroes of your $Y$. Then $$ Lie (G_Y) = \{ X \in {\mathfrak{gl}}(V) | X(I(Y))\subseteq I(Y)\}. $$ Now you know that $I(Y)$ is homogeneous. Pick a finite set of its generators. Let $n$ be the highest degree ...


2

This is an attempt to answer the third question: what else? Let $X$ be a compact space and let $G$ be an affine algebraic group. One can contemplate the following (underived) higher stacks: $BG$: the classifying stack of $G$-torsors. $X_B$: the constant stack associated to $X$. One can consider the higher underived mapping stack $Map(X_B,BG)$, which is ...


2

One historical reason for considering $\ell$-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the $\ell$-adic Tate module of the Jacobian (i.e., the projective limit of $\ell$-power torsion). Furthermore, if such a curve is defined over a subfield ...


2

Mumford (Complex Projective Varieties, section 7) has the following, reasonably simple proof. Let $d$ be the degree of $\mathrm{C}$, $m$ big enough such that $h_{\mathrm{C}}(m)=\mathrm{dim}_{\mathbf{C}} (\mathbf{C}[\mathrm{T}_0,\ldots,\mathrm{T}_n]/\mathrm{I}(\mathrm{C}))_m$ and $md/2>p_a$. Embed $\mathrm{C}$ into $\mathbf{P}^N$ by the degree $m$ Veronese ...


1

P. Bruin and F. Najman have determined the exceptional quadratic points on $X_0(33)$. See Table 8 of https://arxiv.org/pdf/1406.0655.pdf


1

Let $D''$ be a Cartier divisor on $Z$ such that $f^*\mathscr O_Z(D'')\simeq \mathscr O_X(mD')$. Then by the projection formula $\mathscr O_Z(D'')\simeq f_*\mathscr O_X(mD')$ (since $f_*\mathscr O_X\simeq \mathscr O_Z$). On the other hand, from the construction we see that $\mathscr O_Z(D'')|_{Z_0}\simeq f_*\mathscr O_X(mD')|_{Z_0}\simeq \mathscr O_Z(mD)|_{...


1

It is even finitely presented See Lemma 14.1.27 of the book Derived Categories (also available at the arXiv at https://arxiv.org/abs/1610.09640).


1

Let me first reformulate this question slightly. Set $S = \text{Spec}(R)$, denote $\eta \in S$ the generic point (with residue field $k$) and denote $s \in S$ the closed point (with residue field $L$). We may assume $Z$ is not only irreducible but also reduced as the question is about points. I assume that $A \subset k[X_0, \ldots, X_n]$ is exactly the set ...


1

Maybe what follows should be modified depending on what is exactly assumed in the question; nevertheless, let me try. We fix $m$ positive integer large enough so that $h_C(m)=dim H^0(C,\mathcal{O}_C(m))$. We have to show that $dim H^0(C,\mathcal{O}_C(m)) \leq deg(D)m +1$. Let $H$ be a degree $m$ hypersurface of $\mathbb{P}^n$ intersecting $C$ in finitely ...


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