# Lorenzo Mantovani

## Plane Algebraic Curves

### (Mat 568, Spring Semester 2019)

#### Introduction

In the first part of the seminar we will quickly review the basics of projective geometry over a field $K$. This will be essential for the study of algebraic curves, since it will unlock many statements that would not be true if stated in the usual affine plane, where there exist parallel lines. Indirectly this section is about the easiest example of algebraic curves (lines, and pencils of lines).

In the second part: curves, equations, pencils of curves. Detailed study of conics ad pencils of conics. Singular and inflection points and how to detect them. Another example: rational curves. How to construct new curves out of given ones: polars, hessians, and tangential curves.

In the third part: intersections of curves via resultants...

In the fourth part: local study of curves by means of power series...

Birational geometry of curves and projective curves in general

Linear systems on curves and Riemann-Roch...

#### References

The main references for this course are the following books. What we cover during the course is a strict subset of the content of such books, but we often go into some more details and computations.

• R. J. Walker, Algebraic Curves, Princeton university Press, 1950
• A. Seidenberg, Elements of the theory of algebraic curves, Addison-Wesley, 1968
• G. Fischer, Plane algebraic curves, American Matheatical Society, 2001
• E. Brieskorn, H. Knörrer, Plane Algebraic Curves, Birkhäuser, 1986

For the more intrepid student I suggest the following book, which assumes a bit of confidence with commutative algebra.

• W. Fulton, Algebraic Curves, Link

For the people who had already a course of algebraic geometry, I suggest comparing the content of the lectures with chapter 7 of the following book:

• Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press.

#### Program

##### Week 1-2: Rudiments of projective geometry.

Introduction to the geometry of the complete projective space $\mathbb S(V)$, which is the collection of sub-vector spaces of a given finite dimensional vector space $V$ over the field $K$. Definition of the projective space of points $\mathbb P(V)$ and of hyperplanes $\mathbb I(V)$ of $\mathbb S(V)$. Definition of projective subspaces $\mathbb S(W)$ and of projective stars $\mathbb S(V/W)$ associated with a sub-vector space $W\subset V$. The dual projective space and the duality principle. Grassmann dimension formula. Projective coordinate systems and their duals. Homogeneous coordinates of a point $P\in \mathbb P(V)$, homogeneous equations for an hyperplane $\pi \in \mathbb I(V)$ and the relation with its homogeneous coordinates as a point in $\pi \in \mathbb P(V^\ast)$. Homogeneous equations of projective subspaces of $\mathbb P(V)$. Maps between projective spaces and their matrix representation dual maps and their matrix representation. Definition of affine spaces as the complement of a hyperplane in $\mathbb P(V)$, cellular decomposition of a projective space into affine spaces. Interpretation of non-homogeneous linear systems of equations in terms of projective spaces.

##### Week 3: Algebraic curves.

Polynomial rings and homogeneous polynomial rings. Plane curves and hypersurfaces. Hilbert's Nullstellensatz, and how to reconstruct an equation of a curve from its points. Divisors and their intersection with lines. Linear systems of curves/divisors. As an example we review projective conics and their classification up to projectivity over some special field (e.g. quadratically closed fields, real closed fields, finite fields (when time permits). As an example of linear system, we classify pencils (linear systems of dimension one) of conics over an algebraically closed field.

##### Week 5: Intersections of curves

Resultants of polynomials, and their use to detect if two given polynomials $f,g\in K[T]$ have a common root. Various properties of resultants and various ways of writing them. Resultants of homogeneous polynomials are homogeneous.

Bezout Theorem:

• definition of intersection multiplicity $m_P(C,C')$ in a point $P$ between two arbitrary curves $C,C'$ having no common component;
• $C\cdot C'$is a divisor of degree $\mathrm{deg}(C)\mathrm{deg}(C')$;
• $m_P(C,C')\geq m_p(C)m_P(C')$ and equality holds iff the two curves have distinct tangents in $P$.

Various applications of Bezout theorem: estimate of the number of tangents to a curve $C$ from a given point $Q$; estimate of the number of inflexion points; estimate on the number of singularities; a condition of rationality given by degree and number of singular points.

Other applications of resultants: how to find the equation of a rational plane curve given a parametrization.

##### Week 6-7: Local study of curves.

Rings of power series in one indeterminate $K[[T]]$. The order of a power series, the induced valuation and the corresponding non-archimedean norm. Topological properties of power series rings: separateness, completeness, (non-)connectedness. The substitution operation, its properties, and the interpretation as $K$-algebra continuous endomorphism of $K[[T]]$. Hensel lemma about the lifting of factorizations and simple zeros of polynomials. The ring $K[[T^{1/\infty}]]$ of fractional (Piuseux) power series and its fraction field $K((T^{1/\infty}))$. The Newton-Puiseux theorem on the algebraic closeness of $K((T^{1/\infty}))$ in characteristic zero over algebraically closed base fields $K$. The Newton polygon and the explicit determination of roots of polynomials with coefficients in $K((T^{1/\infty}))$.