MAT 615 Course Webpage Topics in Algebraic Geometry: Spaces of Curves on Algebraic Varieties Tu Th 2:30PM — 3:50PM, Physics P—122 Fall 2018

Course Announcements Announcements about the course will be posted here. Please check the site regularly for announcements (which will also be given in lecture and/or in recitation).

Course Description This course focuses on the most classical parameter spaces / moduli spaces in algebraic geometry: moduli spaces of curves and parameter spaces of curves in a fixed algebraic variety. In addition to introducing the foundational techiniques in algebraic geometry for working with such moduli spaces (Artin's algebraization theorems, Grothendieck-Riemann-Roch, ampleness criteria, connectedness theorems, etc.), we will focus on specific applications such as the following:

• construction and properties of Gromov-Witten invariants, including "enumerativity" of these invariants,
• the proof of the Koll'ar-Miyaoka-Mori conjecture,
• the negative answer to Serre's question about rational points of O-acyclic varieties over function fields of curves,
• the positive proof of Serre's "Conjecture II" about rational points of homogeneous varieties over function fields of surfaces and relatied fields, and
• symplectic deformation invariance of rational surfaces (complex dimension 2) on complex projective manifolds in terms of Gromov-Witten invariants.

Prerequisites Students should have passed the graduate algebra sequence. A basic understanding of the language of modern algebraic geometry will also be essential.

Text There is no required textbook. For the first few weeks, I will present foundational theorems in the area (Mori's proof of Hartshorne's conjecture, the Kontsevich recursion formulas for genus 0 curves in the plane, boundedness of Fano manifolds, etc.). For general background, I recommend the following two excellent textbooks.

• János Kollár, Rational Curves on Algebraic Varieties.
• Olivier Debarre, Higher-dimensional Algebraic Geometry.

• Tom Graber, Joe Harris and Jason Starr, Families of rationally connected varieties.
• A. J. de Jong and Jason Starr, Every rationally connected variety over the function field of a curve has a rational point.
• Brendan Hassett and Yuri Tschinkel, Weak approximation over function fields.
• Tom Graber, Joe Harris, Barry Mazur and Jason Starr, Rational connectivity and sections of families over curves.
• János Kollár, A conjecture of Ax and degenerations of Fano varieties.
• Amit Hogadi and Chenyang Xu, Degenerations of rationally connected varieties.
• A. J. de Jong and Jason Starr, Higher Fano manifolds and rational surfaces.
• Carolina Araujo and Ana-Maria Castravet, Polarized minimal families of rational curves and higher Fano manifolds.
• A. J. de Jong and Jason Starr, Almost proper GIT-stacks and discriminant avoidance.
• A. J. de Jong, Xuhua He and Jason Starr, Families of rationally simply connected varieties over surfaces and torsors for semisimple groups.
• Jason Starr, Rational points of rationally simply connected varieties.
• Brendan Hassett, Weak approximation and rationally connected varieties over function fields of curves.

Lectures The instructor for this course is Jason Starr. All instruction will occur in lectures. The tentative schedule is in the syllabus.

Lecture is held Tuesdays and Thursdays, 2:30 PM — 3:50PM in Physics P—122.

Office Hours Office hours are to be announced.

Grading System The grading system will be discussed at the first class meeting.

Americans with Disabilities Act.
If you have a physical, psychological, medical or learning disability that may impact your course work, please contact the Student Accessibility Support Center, ECC (Educational Communications Center) Building, Room 128, (631)632-6748. They will determine with you what accommodations, if any, are necessary and appropriate. All information and documentation is confidential.

Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and the Student Accessibility Support Center. For procedures and information go to the following website: https://ehs.stonybrook.edu/programs/fire-safety/emergency-evacuation/evacuation-guide-people-physical-disabilities and search Fire Safety and Evacuation and Disabilities.

Academic Integrity Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. Faculty in the Health Sciences Center (School of Health Technology & Management, Nursing, Social Welfare, Dental Medicine) and School of Medicine are required to follow their school-specific procedures. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at https://www.stonybrook.edu/commcms/academic_integrity/index.html .

Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of University Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn. Faculty in the HSC Schools and the School of Medicine are required to follow their school-specific procedures.

Back to my home page.