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Tesi etd-06252021-171335


Tipo di tesi
Tesi di dottorato di ricerca
Autore
CONTI, FEDERICO CESARE GIORGIO
URN
etd-06252021-171335
Titolo
Surfaces close to the Severi lines
Settore scientifico disciplinare
MAT/03
Corso di studi
MATEMATICA
Relatori
tutor Prof.ssa Pardini, Rita
Parole chiave
  • geography of surfaces
  • maximal Albanese dimension
  • Severi type inequalities
  • surfaces close to the Severi lines
  • surfaces of general type
Data inizio appello
02/07/2021
Consultabilità
Completa
Riassunto
In this Thesis we study surfaces of general type with maximal Albanese dimension for which the quantity $K_X^2-4\chi(\O_X)-4(q-2)$ vanishes or is "small", that is surfaces close to the Severi lines.
Over the complex numbers, it is known that a surface $X$, provided that $K_X^2<\frac{9}{2}\chi(\O_X)$, has to satisfy the inequality $K_X^2-4\chi(\O_X)-4(q-2)\geq 0$. We give a constructive and complete classification of surfaces for which equality holds: these are surfaces whose canonical model is a double cover of an Abelian surface ($q=2$) or of a product elliptic surface ($q\geq 3$) branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice in the latter case.

We also prove, in the same hypothesis, that a surface $X$ with $K_X^2\neq 4\chi(\O_X)+4(q-2)$ satisfies $K_X^2\geq 4\chi(\O_X)+8(q-2)$ and we give a characterization of surfaces for which the equality holds. These are surfaces whose canonical model is a double cover of an isotrivial smooth elliptic surface fibration branched over an ample divisor with at most negligible singularities whose intersection with the elliptic fibre is $4$.

Because these results are intimately related to theory of double covers, we see that their proof extend almost step by step to the case of any algebraically closed field of characteristic different from $2$.
We also give some partial results over algebraically closed fields of characteristic $2$ after a study of double covers in that case.
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