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For sure if the variety is a complete intersection then the normal bundle splits. I am afraid that the other way round is not true. IMO it should imply just being locally complete intersection, and a …

All smooth genus four curves have two $g^1_3$. Since the unviersal picard fibration over the moduli space of stable genus four curves is proper, yes you can assume that the generic fiber is hyperellip …

You can probaly describe your space as some kind of fibration in $\mathcal{M}_{0,m}$ over the dual linear system $\mathbb{P}^{n*}$, via Kapranov's blow-up construction of $\overline{\mathcal{M}_{0,m}} …

$\frac{1}{2}(d-1)(d-2)$ is the genus of a smooth plane curve of degree $d$. If you project from $P^3$ to $P^2$ off a point not contained in $C$ you can always get a plane curve of the same degree with …

I have always had problems in understanding what people mean when they say "the generalized Theorem of ...". For instance, I seem to understand that there are a few different generazlized Noether-Lefs …

Just a remark: the Pfaffian locus is not quite the $\mathcal{C}_{14}$ divisor, but rather a constructible dense subset of $\mathcal{C}_{14}$, see https://link.springer.com/article/10.1007/s00208-018 …

Let $p : W \rightarrow S$ be a smooth morphism of algebraic varieties over $\mathbb{C}$. Is it always true that: $$p^*H^{p,n-p}(S)=p^*H^n(S,\mathbb{C})\cap H^{p,n-p}(W),$$ whenever $n\geq p$? If not …

What is the standard reference for a definition , valid in all characteristics, of the residue in a point of a rational differential form on a curve?

Suppose $X$ is a smooth projective complete intersection contained in the product $\mathbb{P}^n \times \mathbb{P}^m$, and call $X_n$ and $X_m$ the images of $X$ inside $\mathbb{P}^n$ and $\mathbb{P}^m …

Is the moduli space of Prym curves (curves $C$ with square root of $\mathcal{O}_C$, compactified via admissible covers - by Beauville) of a given genus $g$ normal? Why?

I am maybe misunderstanding, but I think that ypur example is fulfilled by a couple of curves of gwnus bigger than 3, one hyperelliptic and the other not. If you consider the canonical divisor, it is …

What is the state-of-the-art of the proof/counterexamples of the Hasse principle for high-dimensional hypersurfaces (say 3-folds or more)?

as long as $f$ passes to the quotient (i.e. sends orbits on orbits) it has the required property. Moreover, as Misha remarked, the semistable locus of the first quotient should be sent to the semistab …

$\mathbb{P}^3$ compactifies the moduli space of genus 2 curves with level 3 structure and the choice of an odd theta characteristic.

I like very much Mumford's "abelian varieties", which has the advantage to work over any characteristic. enjoy!