**Algebraic Curves**

V5D2 - Selected Topics in Topology

Filip Misev

MPIM, office 222

fmisev@mpim-bonn.mpg.de

**Time and place:**

Tuesday 14:15 - 16:00, Endenicher Allee 60, Neubau, Seminarraum N 0.003.

Lecture began: 2. April 2019.

Last lecture: 9. July 2019.

**Oral exam:**

Monday, 29. July 2019 and Tuesday, 30. July 2019

Place: Endenicher Allee 60, Neubau, Seminarraum N 0.007.

Duration: 20 minutes.

Second exam date: Tuesday, 24. September 2019

Place: Endenicher Allee 60, Seminarraum 1.007 (main building).

Duration: 20 minutes.

**Topics covered:**

*2. April:* Bézout's theorem.

Definition of algebraic curves in **C**^{2}, degree, resultant of two polynomials. First examples of algebraic curves. How to geometrically "see" the degree of a curve by counting intersection points with a line. Examples of pairs of curves intersecting. Proof of (weak version of) Bézout's theorem for curves in **C**^{2}. Excursion to polynomials over the real numbers (see exercises).

[Brieskorn-Knörrer, section 6.1] [Ghys, p.11]

*9. April:* Projective curves, intersection multiplicity.

Projective space: definition of **P**^{n}(**R**) and **P**^{n}(**C**), coordinate charts, definition of projective algebraic curves. Homogeneous polynomials, homogenisation of a polynomial. Correspondence between curves in **C**^{2} and curves in **P**^{2}(**C**). Definition of intersection multiplicity *i*_{p}(C,C') at a point p ∈ C ∩ C', using the resultant. Statement of Bézout's theorem for projective curves. Examples: two parallel lines meet at infinity, a parabola "closes up" to a sphere when seen in the projective plane, intersection multiplicity between a parabola and a tangent line. Application of Bézout's theorem: Pascal's "hexagrammum mysticum" (proof in the next lecture).

[Fulton, chapters 4 and 5]

*16. April:* Applications of Bézout's theorem, genus of a smooth projective curve.

Lemma (corollary of Bézout's theorem): C_{1}, C_{2} curves of degree d intersecting in d^{2} points, C' an irreducible curve of degree k < d passing through kd of the points C_{1} ∩ C_{2}, then there exists a curve C'' of degree d − k which goes through the remaining d(d − k) intersection points.

Application: Pascal's theorem: The intersection points of the three pairs of opposite sides of a hexagon inscribed in an ellipse are collinear.

The space of projective plane curves of fixed degree is itself a projective space. Genus of a smooth projective plane curve of given degree: 2g = (d − 1)(d − 2).

*23. April:* Linear systems, genus-degree formula.

Definition of linear systems; curves through a given set of points. Some details on the proof of the genus-degree formula (compare exercise 11). Chow's theorem (without proof): compact Riemann surfaces in **P**^{2}(**C**) are algebraic curves.

*30. April:* The link of a singularity, Newton polygon, Puiseux' theorem.

How to see singular points on algebraic curves, even without a magnifying glass, by deforming the equation. Beginning to study the local topology of an algebraic curve in the neighbourhood of a singular point. Examples: the cusp y^{2} = x^{3} (trefoil knot), also seen as a perturbation of the double circle y^{2} = 0. Further examples: (y^{2} − x^{3})^{2} − x^{7} and (y^{2} − x^{3})^{2} − x^{5}y as perturbations of the cusp. Definition of weighted-homogeneous polynomials, Newton polygon of a polynomial f(x, y). Statement of Puiseux' theorem: every algebraic curve C ⊂ **C**^{2} is locally a union of finitely many branches of the form t ↦ (t^{m}, g(t)), where m is a positive integer and g is a power series. Definition of Newton series, Puiseux series, Newton pairs, Puiseux pairs.

[Wall, chapter 2]

*7. May:* Newton's algorithm.

Description of Newton's algorithm to compute Newton/Puiseux series for the branches of f(x, y) = 0 at (0, 0), example: y^{2} − x^{3} = 0 and (y^{2} − x^{3})^{2} − 4x^{5}y = 0, proof of boundedness of denominators of fractional powers of x in the series produced by Newton's algorithm.

[Brieskorn-Knörrer, section 8.3]

*14. May:* Knots, links, Seifert surfaces.

Brief introduction to knot theory: knots and links in S^{3}, isotopy, diagrams, positive and negative crossings, Seifert surfaces. Every link has a Seifert surface: Frankl-Pontrjagin's proof and Seifert algorithm. Linking numbers: various definitions.

[Rolfsen, chapter 5 (also chapters 1, 2)] [Frankl-Pontrjagin: Ein Knotensatz mit Anwendung auf die Dimensionstheorie, Math. Ann. 102 (1930), pp. 785-789 (original article in German)] [Seifert: Über das Geschlecht von Knoten, Math. Ann. 110 (1934), pp. 571-592 (original article in German).]

*21. May:* Cable knots.

Torus knots, meridian and (preferred) longitude of an embedded torus in S^{3}, definition of the (a, b)-cable of a knot K, denoted K_{(a, b)}. Description of the knot associated to a branch of an algebraic curve near a singular point in terms of its Newton-Puiseux series: it is an iterated cable of the unknot, and the cabling coefficients (p_{i}, α_{i}) can be computed from the Newton pairs (p_{i}, q_{i}) using the recursive formula α_{1} = q_{1}, α_{i+1} = q_{i+1} + p_{i+1}p_{i}α_{i}.

[Eisenbud-Neumann, Appendix to Chapter I: Algebraic links]

*28. May:* Conical structure theorem, positive braids, Milnor's fibration.

Conical structure theorem: If C ⊂ **C**^{2} is an algebraic curve passing through (0, 0), there exists ε_{0} such that (1) K_{ε} := C ∩ S_{ε}^{3} is a link which is independent of ε ≤ ε_{0} and (2) near the point (0, 0), the curve C is locally embedded as the cone over the link: (B_{ε}^{4}, C ∩ B_{ε}^{4}) ≅ (B_{ε}^{4}, Cone(K_{ε})), for ε ≤ ε_{0}. Definition of the braid group on n strands as the fundamental group of the space of polynomials with n distinct roots; definition of positive braids.

Theorem: Links of singularities are positive braids. Proof sketch using the description of the link of a singularity as an iterated cable of the unknot. Example of a singularity with two branches whose Newton series coincide to the first approximation y^{(1)} = x^{3/2}(1 + x^{1/4}) and y^{(2)} = x^{3/2}(1 + x^{1/6}): the corresponding knots are cables of the T(2,3) torus knot which lie on concentric tori of distinct radii.

Definition of locally trivial fibre bundles, definition of fibred links. Statement of Milnor's fibration theorem (links of singularities are fibred: the argument map of the defining polynomial is a locally trivial fibration). First examples and illustrations: the Hopf link, elastic bands.

[Wall, chapter 5] [Milnor]

*4. June:* Repetition session.

Seifert surface of genus zero for the link in exercise 22, Seifert's algorithm applied to the same link gives a surface of genus two. Repetition/more precise discussion of the definition of genus as the number of disjoint "broken glasses" that can be embedded in a given surface.

Discussion on how/when the Newton algorithm stops: sequence of multiplicities m_{i} (lowest exponents of y alone appearing in f_{i}) decreases. The p_{i} can become equal to one and go back to numbers strictly bigger than one, as it happens for instance in exercise 21 (b).

Links of singularities are iterated cables of the unknot: details on why the link of a singularity is an iterated cable (comparison of the radii of the tori on which the cables lie). Illustrated with the example y = x^{3/2} + x^{7/4}. Meridian and longitude; drawing of the (2, 13)-cable of the trefoil knot T(2, 3).

*25. June:* Fibration and monodromy of algebraic knots.

Theorem: Algebraic knots are fibred. Proof by showing that cables of fibred knots are fibred. Monodromy of a fibred knot as the glueing map of a mapping torus and as the flow of a vector field. Example: the monodromy of the torus knot T(a, b) = S^{3} ∩ {y^{a} = x^{b}} is periodic, using the flow (x, y) ↦ (ζ_{t}^{a} x, ζ_{t}^{b} y), where ζ_{t} = e^{2πit/ab}.

[Wall, chapter 6] [Milnor]

*2. July:* Seifert form, Alexander polynomial, homological monodromy.

Homology of a surface. Definition of the Seifert form associated to a Seifert surface, definition of the Alexander polynomial of a knot. Theorem: The Alexander polynomial of a fibred knot equals the characteristic polynomial of its homological monodromy (proof using the monodromy flow). Consequence: the Alexander polynomial of a fibred knot is monic and of degree 2g, where g is the genus of its fibre surface.

[Rolfsen, 8.C.5.] [N. Saveliev, Lectures on the topology of 3-manifolds, De Gruyter Textbook, 2012, ISBN: 978-3-11-025035-0, chapter 8 (Fibered Knots)]

*9. July:* Blowing up, resolution of singularities and the monodromy.

Idea of blowing up: polar coordinates. Real oriented blow-up, real unoriented blow-up, complex blow-up. Example: cusp. Description of the Milnor fibre and the monodromy of a singularity based on a resolution tree with multiplicities obtained by a sequence of blow-ups resulting in normal crossing divisors. The monodromy decomposes into periodic pieces which are glued together along cylinders, where the periods are given by the multiplicities of the exceptional divisors and the cylinders arise near a crossing between two exceptional divisors.

[A'Campo: La fonction zęta d'une monodromie, Commentarii Mathematici Helvetici 50 (1975), pp. 233-248 (original article in French; see paragraph 2 therein)]

**Exercises:**

I give exercises throughout the lecture which are meant to help follow^{1} the class. They are optional and will not be graded.

**Description:**

An algebraic curve in **C**^{2} is the set of zeros of a polynomial in two complex variables, for example {(x, y) ∈ **C**^{2} : y^{4} − 2x^{3}y^{2} − 4x^{5}y + x^{6} − x^{7} = 0}. The subject is very rich and combines several fields of mathematics, from algebra to geometry, topology and number theory. Our goal is to highlight some of these many faces of algebraic curves. The main focus of the lecture will be on the embedded topology of an algebraic curve near (as well as far from) its singular points, which is described by certain knots and links in the three-dimensional sphere. We will study the properties of such knots and links and use them to classify algebraic curves. In parallel, we will also take the algebraic side of the story into account, including invariants such as intersection numbers, degree and multiplicity, and get a glimpse of algebraic geometry.

**Prerequisites:**

No prerequisites beyond the typical mathematics bachelor contents.

**Further reading:**

- J. Milnor:
*Singular points of complex hypersurfaces*

Princeton University Press, 1968. - D. Eisenbud, W. Neumann:
*Three-dimensional link theory and invariants of plane curve singularities*

Annals of Mathematics Studies, 110. Princeton University Press, Princeton, NJ, 1985. - E. Brieskorn, H. Knörrer:
*Ebene algebraische Kurven*

Birkhäuser, 1988. (Translated by J. Stillwell:*Plane algebraic curves*, Birkhäuser 2012.) - W. Fulton:
*Algebraic curves. An introduction to algebraic geometry*

Reprint of 1969 original, Addison-Wesley Publishing Company, Redwood City, CA, 1989. Free pdf copy available on the author's webpage. - E. Ghys:
*A singular mathematical promenade*

Ecole Normale Supérieure, 2017. Free pdf copy available on the author's webpage. - C.T.C. Wall:
*Singular points of plane curves*

London Mathematical Society, Student Texts 63, Cambridge University Press, 2004. - D. Rolfsen:
*Knots and Links*

AMS Chelsea Publishing, 1976.

^{1} (or, getting distracted from)

9. August 2019