9
votes
What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?
(1) There is the following indirect explanation:
For a generic curve neither of these phenomena happen - the discriminant has no repeated roots and the branch points all have ramification index two. ...
9
votes
Accepted
What relationship is there between repeated roots of discriminants and orders of roots of the original polynomials?
$\def\P{{\mathbb{P}}}
\def\A{{\mathbb{A}}}
\newcommand{\O}{\mathcal{O}}
\DeclareMathOperator{\Disc}{Disc}$I think the story goes like this. The multiplicity of a zero of the discriminant counts ...
4
votes
Intersection complex of genus-zero curves?
Not a complete answer, too long for a comment. I think you'd find it useful to think about these things in terms of Hassett's moduli spaces of weighted pointed curves. Here's the brief version. Fix a ...
3
votes
Accepted
Lifting of quadrics containing a curve
1-normality is sufficient, i.e. it suffices that $H^0(\mathbb P^r, \mathcal O_{\mathbb P^r}(1))\to H^0(C, \mathcal O_C(1))$ is surjective.
Indeed, choose class $q \in H^0(\mathbb P^r, \mathcal O_{\...
2
votes
Motivation for zeta function of an algebraic variety
Here's the simple motivation going back to the beginning with Riemann of the cottage industry of zeta functions.
The Euler product for the Riemann zeta function is
$$\zeta(s) = \prod_p \frac{1}{1-p^{-...
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