__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)

**(UNIFE, October - December 2022)**

__- Algebraic Geometry II____ __** - 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)

__(IMPA - UFF, Escola Transguanabara de Geometria Algébrica, Ph.D. Course, February 2020)__

- Introduction to Toric Geometry and Cox Rings- Introduction to Toric Geometry and Cox Rings

**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)

__(UFF, March - July 2017)__

- Calculus 1A:- Calculus 1A:

__Differential calculus of functions of one variable__

__(UFF, November 2015 - March 2016, April - August 2016, August - December 2016, August - December 2017, August - December 2018)__

**- Calculus 2B:**__Differential calculus of functions of several variables____(IMPA, Ph.D. Course, January - February 2015)__

- Topics in Algebraic Geometry: Cox Rings- Topics in Algebraic Geometry: Cox Rings

**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.

**349.**

Room:

Room:

**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*, Jour. Alg. Geom. 2, 1993, 239-262.

*M0,n*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.

**228.**

Room:

Room:

**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.