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Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n.$ Does $X$ contain a smooth surface $S$ with $c_1(T_S)^2>2c_2(T_S)$? For $d<<\sqrt{n}$ the answer is yes, as $X$ will contain a p …

By a result of Mori, we know that there are only finitely many deformation classes of dimension $n$ smooth projective Fano varieties, for any $n$. For curves we only have $\mathbb{P}^1$, while for sur …

A cubic threefold is a smooth degree $3$ hypersurface in $\mathbb{P}^4$. Is there a cubic threefold $X$ over any field $k$ (possibly of positive characteristic) and an odd degree rational map $\mathbb …

Let $X$ and $Y$ be two algebraic varieties over $\mathbb{C}$. I am interested in the cohomology of $\operatorname{Hom}(X_{top},Y_{top}),$ where I am taking homomorphisms between the underlying topolog …

(In characteristic zero) Flatness implies the other two definitions; integrability and formal lifting are very weak conditions (in fact if I haven't made a mistake I think this notion of integrability …

Let's say I have a smooth irreducible subvariety $X$ of $\mathbb{CP}^n$ with some fixed Hilbert polynomial. What are the best bounds known for the sum of the Betti numbers of $X$? That such a bound ex …

No, take $f=b^4-a^3c$ and $g=ac^3.$ Now the variety $V(b^4-a^3c,y^2-ac^3)$ being reducible is equivalent to the ideal $(b^4-a^3c,y^2-ac^3)$ being prime. But we have $(ya-b^2c)(ya+b^2c)=y^2a^2-b^4c^2=a …

All perfect complexes on an open $U$ of a smooth variety $X$ extend to perfect complexes on $X$. In fact, I believe a much more general statement is true: For an open substack $U$ of a separated Noeth …

Some comments: It is not necessarily true that chiral algebras are essentially conformal vertex algebras, as chiral algebras are allowed to vary over the curve in a way that vertex algebras are not. …

Is there an example of a non-lci morphism $X\rightarrow Y$ for which the entire cotangent complex (or just Andre-Quillen cohomology) can be explicitly computed? I believe it is a theorem of Avramov th …

Really this is mostly just consolidating what has been (implicitly) said in the comments and cleaning it up a bit (e.g. using the Chow ring instead of singular cohomology), but might as well make it a …

Let $X$ be a variety of arbitrary dimension, let $H$ denote the main component of the Hilbert scheme of points of $X$ (i.e, the closure of locus of reduced subschemes), and let $Z$ be the reduced fibe …

As pointed out in the comments, this is Theorem 6.3.1.2 of Hilburn-Raskin. (It certainly was known much earlier, but I'm not sure what to give as a reference.) Their proof is stated quite elegantly, i …

The set $S$ gives rise to a subscheme (which let's also denote by $S$) of $\mathbb{P}^1_{\mathbb{Z}},$ because relatively a prime pair $(x,y)$ corresponds to a section of $\mathbb{P}^1_{\mathbb{Z}}\ri …

Consider the distribution of the number of $\mathbb{F}_q$ points as I range over smooth projective curves of genus $g$ (defined over $\mathbb{F}_q$). If $q\gg g,$ the Hasse-Weil bounds give me a lot o …