Combinatorial aspects of sequences of blow-ups

• Autores: Daniel Camazón Portela
• Directores de la Tesis: Antonio Campillo López (dir. tes.) , Santiago Encinas Carrión (codir. tes.)
• Lectura: En la Universidad de Valladolid ( España ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: Evelia García Barroso (presid.) , Ann Lemahieu (secret.) , F. Monserrat Delpalillo (voc.)
• Enlaces
  • Tesis en acceso abierto en: UVADOC
• Resumen

  • We study sequences of blow-ups at smooth centers $Z_{s}\xrightarrow{\pi_{s}} Z_{s-1}\xrightarrow{\pi_{s-1}}\cdot\cdot\cdot\xrightarrow{\pi_{2}} Z_{1}\xrightarrow{\pi_{1}} Z_{0}$ and the associated sequential morphism $\pi: Z_{s}\rightarrow Z_{0}$. To this end, we introduce the key concept of a final divisor, that is, an irreducible exceptional component for which there exists an open set $U_{i}$ on $Z_{i}$, with $E_{i}^{i}\subset U_{i}$, such that the restriction of the composition $\pi_{i+1}\circ\pi_{i+2}\circ...\circ\pi_{s-1}\circ\pi_{s}\vert_{U_{i}}$ defines an isomorphism. Furthermore, we study the admissible proximity relations between two final divisors with non empty intersection.\\ In the case of sequences of point blow-ups in arbitrary dimension and the corresponding sequential morphisms, we define two equivalence relations: the algebraic equivalence and the combinatorial equivalence, which allow us to classify them. By proving a result that characterizes final divisors in terms of some relations defined over the Chow group of zero-cycles of its sky, we are able to recover the sequence of blow-ups, modulo algebraic equivalence, from the associated sequential morphism. As a result, we establish a connection between the corresponding algebraic and combinatorial equivalence classes of these two objects. Moreover, we give an explicit presentation of an algebraic object associated to the sky of a sequence, that is its Chow ring $A^{\bullet}(Z_{s})$, and obtain a surprising result: two sequences of blow-ups of the same length have isomorphic Chow rings. \\ In the case of sequences of point and rational curve blow-ups, we also characterize final divisors by explicitly giving their defining relations over $A_{0}(Z_{s})$, and we introduce an explicit presentation of the Chow ring of its sky $A^{\bullet}(Z_{s})$. By contrast to the case of sequences of point blow-ups, we prove that two sequences of point and rational curve blow-ups may not have isomorphic Chow rings even if they have the same length and proximity relations.