# Philip Engel

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## Registered User

 Name Philip Engel Member for 2 years Seen 4 hours ago Website Location Age
 May15 awarded ● Supporter May14 comment Magic trick based on deep mathematicsHaha, my friend and I barely managed to work through the logic with the audience choosing the numbers 2 and 3! May1 comment Complete curves in $M_g$ and Theta CharacteristicsThanks for the reference, this is exactly what I was looking for. May1 comment Enriques classification of algebraic surfacesOne direct way to show $\kappa(S)=-\infty$ implies $S$ is ruled is to use Iitaka's Conjecture: If $S$ is a surface and $S\rightarrow B$ is a fibration with generic fiber $F$ then $\kappa(S)\geq\kappa(B)+\kappa(F)$. Applying this to the Albanese fibration solves case (2) above instantaneously, because it implies that $F$ must be rational. This applies the stronger assumption $P_n=0$ for all $n$ rather than $P_{12}=0$ though. I think there really will be no way to get the specific number $12$ without some classification. Apr30 revised Enriques classification of algebraic surfacesadded 23 characters in body; added 47 characters in body Apr30 answered Enriques classification of algebraic surfaces Apr30 asked Complete curves in $M_g$ and Theta Characteristics Apr22 awarded ● Disciplined Mar6 comment Contracting a curve of negative self-intersection on a surfaceThanks for your answer, it partially resolves the case of my second question, by giving criteria for contractibility in the algebraic category in certain cases. I mainly wanted an explicit construction {\it in the analytic category} of the contraction. (The conditions would of course be weaker if we allow the contraction not to be an algebraic surface). Since the proposition above seems to be the best result about contractibility, I assume it is hard then to determine whether the resulting surface is algebraic... Mar1 asked Contracting a curve of negative self-intersection on a surface Feb19 awarded ● Necromancer