DOCSLIB.ORG

Minimal Model Program:Contract Your Bad Curves! Andrea Fanelli,Diletta Martinelli Supervisor:Dr Paolo Cascini

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Minimal Model Program:Contract Your Bad Curves! Andrea Fanelli,Diletta Martinelli Supervisor:Dr Paolo Cascini MINIMAL MODEL PROGRAM:CONTRACT YOUR BAD CURVES! ANDREA FANELLI,DILETTA MARTINELLI SUPERVISOR:DR PAOLO CASCINI INTRODUCTION:OUR GYM SURFACES:ITALIAN VINTAGE TRAINING THREEFOLDS:THE 80’S,FLIPPING PLASTIC SURGERY Costumers: PROJECTIVE VARIETIES over C (geomet- Finally someone who needs an effective workout! In Everyone knows that: the bigger that?!?!? K < 0 ric objects locally defined as zero loci of polynomials, early 20th century, many Italian surfaces put on some you are, the harder is to contract X Problem 3. If we contract a subva- embedded in some projective space). weight, due to fat meals and their mammas’ tiramisu’! bad curves! riety in high codimension, we come They ended up with tons of new curves! up with a new variety where neither Our goal: classify them up to birational equivalence intersection theory makes sense! (we are allowed to cut off some useless and adipose Here’s the new phenomenon in dimension 2: we can K = 0 Solution: There is only one thing subvarieties, keeping essential information on them, X blow up points and obtain infinitely many new sur- to do: FLIP THE CURVE. This is such us their function fields). faces, all birational to the one we started with! a totally different step of the MMP: EXTREMAL μ-1l μ-1l 3 RAYS Main Tool: the canonical bundle KX , which is intrin- 2 starting with X we need to con- + E struct a new variety X (HOW?!?!) sic of our variety and can be useful to detect how the -1 μ l 1 training is going. It is our fat detector! and a birational map μ So now we know our enemy! We P l + l 1 A fat 3-fold 2 l f∶ X⋯ → X Our Training Program (MMP): start with a variety X, 3 start contracting them as for sur- make it sweat and come up with a MINIMAL MODEL faces, BUT... C Xmin. But every blow-up leaves a (fat and superfluous) trace: It took many decades to understand Problem 2. When you contract the cutting off the curve which neg- a rigid (-1)-curve. how to attach the problem for vari- atively intersects KX and replace rays, your variety can become sin- + eties of dimension 3 or bigger! gular! it with a new curve C such that Theorem 2 (Castelnuovo’s Contractibility Criterion). If + Problem 1. (−1)-curves mean noth- Solution: We need to enlarge the (KX+ ⋅ C ) > 0: this is the plastic a non-singular surface S contains a (−1)-curve, we can con- ing on a 3-fold. category we work with ; terminal surgery we need! tract it via a birational morphism φ∶ S → T to obtain another Solution: “rigid” curves are the singularities (the mildest singulari- We expect to end up with a MIN- non-singular surface T . MMP ones which intersect negatively ties you can imagine!) IMAL model (with NEF canonical K And here’s our personalised program for surfaces: the canonical bundle X (extremal bundle) also in this case, but there is rays) ; Mori’s Cone theorem, Well... Fair enough... Even with a priori no reason why the flipped Start which describes the cone of effec- some stretch marks our variety variety should be “skinnier” than X S: non- tive curves on our variety: looks great! OH NO... What is ! So: "Do flips terminate?" singular surface MMP TODAY: BCHM’S LIPOSUCTION FLOWCHART:THE TRAINING TABLE CURVES:TOO SKINNY FOR US! Is there a S ∶= (− ) no c This is the (conjectural) training program for varieties 1 -curve Smin End In 2010, Birkar, Cascini, Hacon and M Kernan devel- Even curves can easily get very complicated! E on S? in arbitrary dimension! C is the category of varieties oped a new revolutionary surgical technique: cooking with terminal singularities. yes up the canonical bundle they can Castelnuovo’s Theorem φ∶ S → T ● construct flips; the contrac- Start S ∶= T tion of E ● come up with a minimal model for a large class of T : nons- X ∈ C ing surface varieties, called of GENERAL TYPE. It requires a lot of work and many techniques, but the yes X ∶= KX nef? End result is outstanding! Xmin After a finite number of steps, the routine ends, since But our work is rather easy for curves: just need to re- no the contraction φ actually simplifies the structure of S Cone and move the bad points! Contraction and we get a MINIMAL surface in perfect shape. FUTURE:FUSION WORKOUT Theorem φ∶ X → Y Theorem 1. Every curve is birationally equivalent to a One can also see that the canonical bundle of Smin has a very interesting property: its intersection with curves What’s left? Tons of varieties still waiting for your help: X is a unique nonsingular projective curve. + dim Y X ∶= X X ∶= Y < yes Mori fiber they really need to loose weight! dim X? End is positive (KSmin is NEF). space Algebraic curves (= Riemann Surfaces) are classified Here’s some open problems: no by their genus g (number of holes) into 3 classes: ● minimal models for many varieties of special type; REFERENCES:THE COACHING STAFF ● Y ∈ C codim E(φ) termination of flips in dimension higher than 4; no g = 0 g = 1 ≥ 2? RATIONAL ( ), ELLIPTIC ( ) AND CURVES ● Debarre, Higher-Dimensional Algebraic Geometry, Springer-Verlag; ● Abundance Conjecture (minimal models are even OF GENERAL TYPE (g ≥ 2) ● Hacon, Kovács, Classification of Higher Dimensional Algebraic Vari- better than expected... they got ripped!); yes eties, Birkhäuser; ● MMP in positive characteristic (this is hardcore!); flip X / X+ ∈ C ● Kollár, Mori, Birational Geometry of Algebraic Varieties, Cambridge ● moduli spaces of varieties of general type (yes, they University Press; " z ● Matsuki, Introduction to the Mori Program, Springer-Verlag, Berlin. are in good shape now... what about a contest?) Y ∈~ C.
Load more

Recommended publications
  • Recent Results in Higher-Dimensional Birational Geometry
    Complex Algebraic Geometry MSRI Publications Volume 28, 1995 Recent Results in Higher-Dimensional Birational Geometry ALESSIO CORTI ct. Abstra This note surveys some recent results on higher-dimensional birational geometry, summarising the views expressed at the conference held at MSRI in November 1992. The topics reviewed include semistable flips, birational theory of Mori fiber spaces, the logarithmic abundance theorem, and effective base point freeness. Contents 1. Introduction 2. Notation, Minimal Models, etc. 3. Semistable Flips 4. Birational theory of Mori fibrations 5. Log abundance 6. Effective base point freeness References 1. Introduction The purpose of this note is to survey some recent results in higher-dimensional birational geometry. A glance to the table of contents may give the reader some idea of the topics that will be treated. I have attempted to give an informal presentation of the main ideas, emphasizing the common grounds, addressing a general audience. In 3, I could not resist discussing some details that perhaps § only the expert will care about, but hopefully will also introduce the non-expert reader to a subtle subject. Perhaps the most significant trend in Mori theory today is the increasing use, more or less explicit, of the logarithmic theory. Let me take this opportunity This work at the Mathematical Sciences Research Institute was supported in part by NSF grant DMS 9022140. 35 36 ALESSIO CORTI to advertise the Utah book [Ko], which contains all the recent software on log minimal models. Our notation is taken from there. I have kept the bibliography to a minimum and made no attempt to give proper credit for many results.
    [Show full text]
  • Introduction to Birational Geometry of Surfaces (Preliminary Version)
    INTRODUCTION TO BIRATIONAL GEOMETRY OF SURFACES (PRELIMINARY VERSION) JER´ EMY´ BLANC (Very) quick introduction Let us recall some classical notions of algebraic geometry that we will need. We recommend to the reader to read the first chapter of Hartshorne [Har77] or another book of introduction to algebraic geometry, like [Rei88] or [Sha94]. We fix a ground field k. All results of the first 4 sections work over any alge- braically closed field, and a few also on non-closed fields. In the last section, the case of all perfect fields will be discussed. For our purpose, the characteristic is not important. n n 0.1. Affine varieties. The affine n-space Ak, or simply A , is the set of n-tuples n of elements of k. Any point x 2 A can be written as x = (x1; : : : ; xn), where x1; : : : ; xn 2 k are the coordinates of x. An algebraic set X ⊂ An is the locus of points satisfying a set of polynomial equations: n X = (x1; : : : ; xn) 2 A f1(x1; : : : ; xn) = ··· = fk(x1; : : : ; xn) = 0 where each fi 2 k[x1; : : : ; xn]. We denote by I(X) ⊂ k[x1; : : : ; xn] the set of polynomials vanishing along X, it is an ideal of k[x1; : : : ; xn], generated by the fi. We denote by k[X] or O(X) the set of algebraic functions X ! k, which is equal to k[x1; : : : ; xn]=I(X). An algebraic set X is said to be irreducible if any writing X = Y [ Z where Y; Z are two algebraic sets implies that Y = X or Z = X.
    [Show full text]
  • MMP) Is One of the Cornerstones in the Clas- Sification Theory of Complex Projective Varieties
    MINIMAL MODELS FOR KÄHLER THREEFOLDS ANDREAS HÖRING AND THOMAS PETERNELL Abstract. Let X be a compact Kähler threefold that is not uniruled. We prove that X has a minimal model. 1. Introduction The minimal model program (MMP) is one of the cornerstones in the clas- sification theory of complex projective varieties. It is fully developed only in dimension 3, despite tremendous recent progress in higher dimensions, in particular by [BCHM10]. In the analytic Kähler situation the basic methods from the MMP, such as the base point free theorem, fail. Nevertheless it seems reasonable to expect that the main results should be true also in this more general context. The goal of this paper is to establish the minimal model program for Kähler threefolds X whose canonical bundle KX is pseudoeffective. To be more specific, we prove the following result: 1.1. Theorem. Let X be a normal Q-factorial compact Kähler threefold with at most terminal singularities. Suppose that KX is pseudoeffective or, equivalently, that X is not uniruled. Then X has a minimal model, i.e. there exists a MMP X 99K X′ such that KX′ is nef. In our context a complex space is said to be Q-factorial if every Weil ⊗m ∗∗ divisor is Q-Cartier and a multiple (KX ) of the canonical sheaf KX is locally free. If X is a projective threefold, Theorem 1.1 was estab- lished by the seminal work of Kawamata, Kollár, Mori, Reid and Shokurov [Mor79, Mor82, Rei83, Kaw84a, Kaw84b, Kol84, Sho85, Mor88, KM92]. Based on the deformation theory of rational curves on smooth threefolds [Kol91b, Kol96], Campana and the second-named author started to inves- tigate the existence of Mori contractions [CP97, Pet98, Pet01].
    [Show full text]
  • MINIMAL MODEL THEORY of NUMERICAL KODAIRA DIMENSION ZERO Contents 1. Introduction 1 2. Preliminaries 3 3. Log Minimal Model Prog
    MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION ZERO YOSHINORI GONGYO Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0. Contents 1. Introduction 1 2. Preliminaries 3 3. Log minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori ¯ber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll¶ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q.
    [Show full text]
  • Birational Geometry of the Moduli Spaces of Curves with One Marked Point
    The Dissertation Committee for David Hay Jensen Certi¯es that this is the approved version of the following dissertation: BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Committee: Sean Keel, Supervisor Daniel Allcock David Ben-Zvi Brendan Hassett David Helm BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT by David Hay Jensen, B.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful¯llment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2010 To Mom, Dad, and Mike Acknowledgments First and foremost, I would like to thank my advisor, Sean Keel. His sug- gestions, perspective, and ideas have served as a constant source of support during my years in Texas. I would also like to thank Gavril Farkas, Joe Harris, Brendan Hassett, David Helm and Eric Katz for several helpful conversations. iv BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Publication No. David Hay Jensen, Ph.D. The University of Texas at Austin, 2010 Supervisor: Sean Keel We construct several rational maps from M g;1 to other varieties for 3 · g · 6. These can be thought of as pointed analogues of known maps admitted by M g. In particular, they contract pointed versions of the much- studied Brill-Noether divisors. As a consequence, we show that a pointed 1 Brill-Noether divisor generates an extremal ray of the cone NE (M g;1) for these speci¯c values of g.
    [Show full text]
  • The Minimal Model Program for B-Log Canonical Divisors and Applications
    THE MINIMAL MODEL PROGRAM FOR B-LOG CANONICAL DIVISORS AND APPLICATIONS DANIEL CHAN, KENNETH CHAN, LOUIS DE THANHOFFER DE VOLCSEY,¨ COLIN INGALLS, KELLY JABBUSCH, SANDOR´ J KOVACS,´ RAJESH KULKARNI, BORIS LERNER, BASIL NANAYAKKARA, SHINNOSUKE OKAWA, AND MICHEL VAN DEN BERGH Abstract. We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work in the setting of the b-log MMP. If we assume that the log MMP terminates, then so does the b- log MMP. Furthermore, the b-log MMP includes both the log MMP and the equivariant MMP as special cases. There are various interesting b-log varieties arising from different objects, including the Brauer pairs, or “non-commutative algebraic varieties which are finite over their centres”. The case of toric Brauer pairs is discussed in further detail. Contents 1. Introduction 1 2. Minimal model theory with boundary b-divisors 4 2.A. Recap on b-divisors 4 2.B. b-discrepancy 8 3. The Minimal Model Program for b-log varieties 11 3.A. Fundamental theorems for b-log varieties 13 4. Toroidal b-log varieties 15 4.A. Snc stratifications 15 4.B. Toric b-log varieties 17 4.C. Toroidal b-log varieties 19 5. Toric Brauer Classes 21 References 24 1. Introduction arXiv:1707.00834v2 [math.AG] 5 Jul 2017 Let k be an algebraically closed field of characteristic zero. Let K be a field, finitely generated over k.
    [Show full text]
  • Effective Bounds for the Number of MMP-Series of a Smooth Threefold
    EFFECTIVE BOUNDS FOR THE NUMBER OF MMP-SERIES OF A SMOOTH THREEFOLD DILETTA MARTINELLI Abstract. We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two. 1. Introduction Establishing the existence of minimal models is one of the first steps towards the birational classification of smooth projective varieties. More- over, starting from dimension three, minimal models are known to be non-unique, leading to some natural questions such as: does a variety admit a finite number of minimal models? And if yes, can we fix some parameters to bound this number? We have to be specific in defining what we mean with minimal mod- els and how we count their number. A marked minimal model of a variety X is a pair (Y,φ), where φ: X 99K Y is a birational map and Y is a minimal model. The number of marked minimal models, up to isomorphism, for a variety of general type is finite [BCHM10, KM87]. When X is not of general type, this is no longer true, [Rei83, Example 6.8]. It is, however, conjectured that the number of minimal models up to isomorphism is always finite and the conjecture is known in the case of threefolds of positive Kodaira dimension [Kaw97]. In [MST16] it is proved that it is possible to bound the number of marked minimal models of a smooth variety of general type in terms of the canonical volume. Moreover, in dimension three Cascini and Tasin [CT18] bounded the volume using the Betti numbers.
    [Show full text]
  • Positivity in Algebraic Geometry I
    Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample.
    [Show full text]
  • Arxiv:1609.05543V2 [Math.AG] 1 Dec 2020 Ewrs Aovreis One Aiis Iersystem Linear Families, Bounded Varieties, Program
    Singularities of linear systems and boundedness of Fano varieties Caucher Birkar Abstract. We study log canonical thresholds (also called global log canonical threshold or α-invariant) of R-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov- Alexeev-Borisov conjecture holds, that is, given a natural number d and a positive real number ǫ, the set of Fano varieties of dimension d with ǫ-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a question of Serre. Next we show that if the log canonical threshold of the anti-canonical system of a Fano variety is at most one, then it is computed by some divisor, answering a question of Tian in this case. Contents 1. Introduction 2 2. Preliminaries 9 2.1. Divisors 9 2.2. Pairs and singularities 10 2.4. Fano pairs 10 2.5. Minimal models, Mori fibre spaces, and MMP 10 2.6. Plt pairs 11 2.8. Bounded families of pairs 12 2.9. Effective birationality and birational boundedness 12 2.12. Complements 12 2.14. From bounds on lc thresholds to boundedness of varieties 13 2.16. Sequences of blowups 13 2.18. Analytic pairs and analytic neighbourhoods of algebraic singularities 14 2.19. Etale´ morphisms 15 2.22. Toric varieties and toric MMP 15 arXiv:1609.05543v2 [math.AG] 1 Dec 2020 2.23. Bounded small modifications 15 2.25. Semi-ample divisors 16 3.
    [Show full text]
  • Automorphisms of Cubic Surfaces Without Points
    AUTOMORPHISMS OF CUBIC SURFACES WITHOUT POINTS CONSTANTIN SHRAMOV Abstract. We classify finite groups acting by birational transformations of a non- trivial Severi–Brauer surface over a field of characteristc zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field K of characteristic zero that has no K-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces. 1. Introduction Given a variety X, it is natural to try to describe finite subgroups of its birational automorphism group Bir(X) in terms of the birational models of X on which these finite subgroups are regularized. In dimension 2, this is sometimes possible due to the well developed Minimal Model Program. In the case of the group of birational automorphisms of the projective plane over an algebraically closed field of characteristic zero, this was done in the work of I. Dolgachev and V. Iskovskikh [DI09]. Over algebraically non-closed fields, some partial results are also known for non-trivial Severi–Brauer surfaces, i.e. del Pezzo surfaces of degree 9 not isomorphic to P2 over the base field. Note that the latter surfaces are exactly the del Pezzo surfaces of degree 9 without points over the base field, see e.g. [Ko16, Theorem 53(2)]. Recall that a birational map Y 99K X defines an embedding of a group Aut(Y ) into Bir(X). We will denote by rk Pic(X)Γ the invariant part of the Picard group Pic(X) with respect to the action of a group Γ, and by µn the cyclic group of order n.
    [Show full text]
  • Birational Geometry of Algebraic Varieties
    Birational geometry of algebraic varieties Caucher Birkar Cambridge University Rome, 2019 Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. Algebraic geometry Algebraic geometry is the study of solutions of systems of polynomial equations and associated geometric structures. Algebraic geometry is an amazingly complex but beautiful subject. It is deeply related to many branches of mathematics but also to mathematical physics, computer science, etc. Let k = Q, or R, or C. Let k[t] = polynomials in variable t with coefficients in k. Given f 2 k[t], we want to find its solutions.
    [Show full text]