Teaching
- Geometry I for mathematics (UNIFE, September 2023 - May 2024, September 2024 - May 2025)
| Lecture notes: P. Ellia, A. Massarenti, Topics in Linear Algebra and Geometry, 2024 |  |
- Algebra and Geometry for electronic and computer engineering (UNIFE, September - December 2024)
- Algebraic Geometry II (UNIFE, October - December 2022)- Algebraic Geometry (UNIFE, February - May 2022, February - May 2023, February - May 2024)
- Geometry (Linear Algebra) for physics (UNIFE, September - November 2020)
- Mathematics for biology (UNIFE, October 2020, October 2021, October 2022)
- Toric Geometry (UNIFE, Ph.D. and Master Course, November - December 2019)
- Mathematics and computer science (UNIFE, February - May 2019, September - December 2019, September - December 2020, September - December 2021, February - May 2023)
- Introduction to Toric Geometry and Cox Rings (IMPA - UFF, Escola Transguanabara de Geometria Algébrica, Ph.D. Course, February 2020)
Topics: convex polyhedral cones, affine toric varieties, fans and toric varieties, toric varieties from polytopes; the orbit-cone correspondence, divisors and line bundles on toric varieties, cohomology of line bundles on toric varieties; ample, nef and movable divisors on toric varieties; the cone of curves of a toric variety; Cox rings and Mori dream spaces. Finite generation of the Cox ring of a toric variety and the Gelfand-Kapranov-Zelevinsky decomposition.
Timetable
References:
D. A. Cox, J. B. Little, H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics (Book 124), American Mathematical Society, 2011.
W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies (Book 131), Princeton University Press, 1993.
J. Hausen, A. Laface, I. Arzhantsev, U. Derenthal, Cox Rings, Cambridge Studies in Advanced Mathematics (Book 144), Cambridge University Press, 2014.
| Lecture notes | |
- Calculus 3A: Integral calculus of functions of several variables (UFF, March - July 2018)
- Calculus 2A: Integral calculus of functions of one variable and ordinary differential equations (UFF, March - July 2017)
- Calculus 1A: Differential calculus of functions of one variable (UFF, March - July 2017)
- Calculus 2B: Differential calculus of functions of several variables (UFF, November 2015 - March 2016, April - August 2016, August - December 2016, August - December 2017, August - December 2018)
- Topics in Algebraic Geometry: Cox Rings (IMPA, Ph.D. Course, January - February 2015)
Topics: Mori Dream Spaces and Cox Rings. Weak Fano and log Fano varieties. Representations of semi-simple Lie algebras and Weyl groups. Secant varieties of rational normal curves. Canonical, terminal and Kawamata log terminal singularities. Log Fano varieties obtained by blowing-up points in projective spaces. Moduli spaces of curves. The moduli space of n-pointed rational curves is not a Mori Dream Space for n > 133.
Timetable: Tuesday, Thursday and Friday, 17:00-18:30.
Room: 349.
References:
A. M. Castravet, J. Tevelev, M0,n is not a Mori Dream Space, to appear on Duke Math. J.
A. M. Castravet, J. Tevelev, Hilbert's 14th problem and Cox rings, Compositio Math, 142, 2006, 1479-1498.
J. Harris, I. Morrison, Moduli of Curves, Vol. 187 of Graduate Texts in Math, Springer-Verlag, New York, 1998.
B. Hassett, Moduli spaces of weighted pointed stable curves, Adv. in Math, 173, 2003, 316-352.
Y. Hu, S. Keel, Mori dream spaces and GIT, Michigan Math. J, 48, 2000, 331-348.
M. Kapranov, Veronese curves and Grothendieck-Knudsen moduli spaces M0,n, Jour. Alg. Geom. 2, 1993, 239-262.
S. Mukai, Counterexample to Hilbert's fourteenth problem for the 3-dimensional additive group, RIMS Preprint n. 1343, Kyoto, 2001.
S. Mukai, Finite generation of the Nagata invariant rings in A-D-E cases, RIMS Preprint n. 1502, Kyoto, 2005.
| Lecture notes | |
- Algebraic Geometry I (IMPA, Ph.D. Course, August - November 2014)
Topics: Affine and projective varieties, Hilbert's nullstellensatz, Zariski's topology, Bézout's theorem. Morphisms and rational maps. Singularities of algebraic varieties. Projective curves. Blow-ups, resolutions of singularities of curves. Veronese and Segre varieties. Grassmannians. Twenty-seven lines in a smooth cubic surface and lines in hypersurfaces of small degree. Theorems on the dimension of the fibers. Secant varieties, rational normal scrolls. Line bundles and divisors. Riemann-Roch theorem for smooth curves.
Timetable: Monday and Wednesday, 9:00-10:30.
Room: 228.
References:
R. Hartshorne, Algebraic Geometry, Berlin, Springer, 1977.
J. Harris, Algebraic Geometry - A First Course, Springer-Verlag, 1992.
I. Shafarevich, Basic Algebraic Geometry 1 - Varieties in projective spaces, Berlin, Springer-Verlag, 1974.
I. Dolgachev, Classical algebraic geometry - A modern view, Cambridge University Press, 2012.