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On the smallest degree of a surface containing a space curve

Margherita Roggero, Paolo Valabrega (1998)

Bollettino dell'Unione Matematica Italiana

Sia C una curva dello spazio di grado D contenuta in una superficie di grado r e non in una di grado r - 1 . Se C è integra, allora r 6 D - 2 - 2 ; questo limite superiore, raggiunto in alcuni casi (cfr. [5]), non vale però per curve arbitrarie (cfr. [?, 3 (iii)]). Ogni curva C dello spazio (anche non ridotta o riducibile) può essere ottenuta come schema degli zero di una sezione non nulla di un opportuno fascio riflessivo F di rango 2. Mediante i fasci riflessivi, siamo in grado di estendere alle curve riducibili...

On the structure of linked 3-folds.

Aldo Biancofiore, Maria Lucia Fania (2001)

Revista Matemática Complutense

The structure of 3-folds in P6 which are generally linked via a complete intersection (f1,f2,f3) to 3-folds in P6 of degree d ≤ 5 is determined. We also give three new examples of smooth 3-folds in P6 of degree 11 and genus 9. These examples are obtained via liaison. The first two are 3-folds linked via a complete intersection (2,3,3) to 3-folds in P6 of degree 7: (i) the hyperquadric fibration over P1 and (ii) the scroll over P2. The third example is Pfaffian linked to a 3-dimensional quadric in...

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

In the current paper we show that the dimension of a family V of irreducible reduced curves in a given ample linear system on a toric surface S over an algebraically closed field is bounded from above by - K S . C + p g ( C ) - 1 , where C denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality 𝚍𝚒𝚖 ( V ) = - K S . C + p g ( C ) - 1 does not imply the nodality of C even if C belongs to the...

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