Ambro, Inversion of Adjunction for non-degenerate hypersurfaces, arXiv: math. Ein, R. Lazarsfeld, M. Ein, M. Yasuda Jet schemes, log discrepancies and inversion of adjunc- tion Invent. Pure Math Amer. Eisenbud and E. Frenkel, Invent. McMullen noticed in [McM03] that the eigenforms for real multiplication over a Hilbert modular surface inter- sected with M2 are invariant under the SL2 R -action. Shimura curves One can think of a Shimura variety as the moduli space of abelian varieties plus an additional endomorphism structure.
We note some consequences of this description: Corollary 3. Corollary 4. Geometry 66 , — On Green and Green-Lazarsfeld conjectures for generic d-gonal curves Marian Aprodu Our main object of study is Koszul cohomology of curves. Koszul cohomology Let X be a complex projective variety. A more geometric description, using spaces of sections in line bundles over Hilbert schemes of points, was given by Voisin in [10]. The aim of the theory is the investigate cases when this fact is revertible.
It was conjectured by Green that this is optimal, i. Green and Lazarsfeld conjectured that this is the best one can do, i. This is actually a fake problem, as one can reduce oneself to verifying the predicted vanishing for a given line bundle: Theorem 1 see [1].
Theorem 2 Hirschowitz-Ramanan-Voisin. The Green conjecture is valid also for curves of non-maximal gonality which are generic in their gonality strata, and this happens for all possible gonalities cf.
Some recent results The breakthrough realized by Voisin in [10], [11] opened the door for further progress in the theory. This fact is used to show the following. Komplexe Algebraische Geometrie Theorem 3 see [3]. Theorem 4 see [3]. Corollary 5 see [3]. Lemma 6 Voisin, [10], [4]. Let Y be an irreducible stable curve, and X be its normalization. The author is a Humboldt Research Fellow. Aprodu, On the vanishing of the higher syzygies of curves, Math. Aprodu, Green-Lazarsfeld gonality conjecture for a generic curve of odd genus, Int.
Notices 63 , — Aprodu, Remarks on syzygies of d-gonal curves, Math. Letters , in press. Aprodu and C. Paris , — Green, Koszul cohomology and the geometry of projective varieties, J. Green and R. Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Hirschowitz and S. Ecole Norm. Tian on log-canonically polarized manifolds.
We have the well known result: Theorem 1. Of the many generalization of the Franchis theorem we recall the classical: Theorem 2. Severi, de Franchis. Partial important results on this conjecture were obtained in [3] and [7]. We have see [7] : Conjecture 1. Iitaka, Severi. Moreover Kani in [5] provided examples showing that m X cannot be polyno- mially bounded. The following conjecture-problem is, in our opinion, the most interesting open problem in the topic.
Conjecture 2. Heier [7] de Franchis problem. We have two steps. Step one: Geometric Lemma. One has Lemma 3. The geometric lemma gives a linear bound for the cardinality of the equivalences classes. The bound is then obtained either by using Kani packing lemma [5] on the lattice H 1 X, Z , or by the interesting reduction mod. This is a polynomial on a. We remark that we do not need the restrictive hypothesis which guarantees the injectivity of the representation of the elements of M X, Y as maps of suitable Hodge structures.
Now we explain the ideas of the proof in surface case. That is X and Y will be minimal surfaces of general type. First we generalize the geometric part of Tanabe to surfaces with pg at least 2 by using appropriate pencil V of 2-forms on Y. Then, by using the fact that the curves are moving in a pencil, and hence they cut this open set, one can reduce to the one-dimensional case. Komplexe Algebraische Geometrie Lemma 5. The result follows on C. Roughly speaking, the transcendental lattice is the complementary of the Neron-Severi group in the second cohomology group of the surface.
The geometric part allows us to estimate the number of maps which are represented by the same couple of elements of the lattice. Observe that, since we are not assuming that X, Y are canonical, the represen- tation of the maps in M X, Y as maps of transcendental Hodge structures is not injective in general.
We need then some additional hypotheses which allow us to give the bound in higher dimensional case following a similar argument. Bandman, G. Estimate of the number of rational mappings from a fixed variety to varieties of general type, Ann. Fourier 47 no. Palermo 36 , Martin-Deschamps and R. France , no. Howard, and A. Sommese, On the theorem of de Franchis, Ann. Pisa 10 , — Kani, Bounds on the number of non-rational subfields of a function field, Invent. Math 85 ,— Kobayashi and T.
Ochiai Meromorphic mappings onto compact complex spaces of general type, Invent. Heier, Effective finiteness theorems for maps between canonically polarized compact com- plex manifolds, ArXiv:math.
London Math. Naranjo and G. Pirola, Bounds of the number of rational maps betweeen varieties of general type, Preprint. Tanabe, A bound for the theorem of de Franchis, Proc. So in some sense these are the surfaces of general type with smallest possible invariants.
Again in both cases there is one 8-dimensional irreducible family. Existence of such surface was proved by Rebecca Barlow, by a complicated quotient construction.
In this talk I present a third approach based on homological algebra following a suggestion of Miles Reid. We study RX as an S-module. The original complex can be obtained from this one by a deformation argument, that is choosing appropriate entries depending on x0 and x1.
The main result so far is the following. We plan to investigate this family and families lying over the complement of V or families with other base locus for 2K with the help of Computer algebra and probabilistic algorithm.
Complex analysis and algebraic geometry, —, de Gruyter, Berlin, Problems in the theory of surfaces and their classi- fication Cortona, , —, Sympos. Carrell asked whether something similar is implied by the existence of a nowhere vanishing holomorphic one form. He proved that this is the case for surfaces, namely if S is a compact complex surface admitting a nowhere vanishing holomor- phic one form, then c1 S 2 and c2 S are zero [Car74]. Therefore we restrict to the case of varieties of non-negative Kodaira dimension.
These considerations naturally lead to the following conjecture. Conjecture 1. Let X be a smooth projective variety of general type. Then X does not admit a nowhere vanishing holomorphic one form. Once we focus on varieties of general type, restricting to the case of minimal varieties is the right thing to do according to a conjecture of Carrell: Conjecture 2 Carrell. If X admits a nowhere vanishing holomorphic one form, then X is minimal.
Remark 1. This is known for surfaces and using the classification of extremal contractions one can easily see that it also holds for threefolds. This was explicitly checked in [LZ03, Lemma 2. An immediate consequence of this conjecture is that a variety of general type does not admit any smooth morphisms onto an abelian variety. We prove Conjecture 1 for smooth minimal varieties and for varieties whose Albanese variety is simple. Assume that either 2.
Using [LZ03, Lemma 2. Next we review the idea of the proof of in the case when X is minimal 2. We want to prove that X does not admit a nowhere vanishing global holomorphic one-form. We prove this via a vanishing theorem. Non compact kahler manifolds, spherical manifolds spherical. Modellannahme f nf sch lerinnen und umschriebene vielecke und euklidische.
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Analytischer Geometrie Kugeloberfl che einer theaterdekoration, die englische lexikon zu formen. Fahrrad Geometrie Neuzeitliche ausgaben der offenen fragen zum inhalt ist vom bildpunkt zum. This is the original version of the class notes, which will not be updated any more.
However, it covers two semesters, and thus contains more material than the new versions above. Version of This version used to be a Bachelor course some time ago. Such a set of zeros is called an affine variety. The affine varieties define a topology on , called the Zariski topology.
As a consequence of Hilbert's basis theorem , each variety can be defined by only finitely many polynomial equations. A variety is called irreducible if it is not the union of two true closed subsets.
It turns out that a variety is irreducible if and only if the polynomials that define it generate a prime ideal of the polynomial ring. The correspondence between varieties and ideals is a central theme of algebraic geometry. One can almost give a dictionary between geometric terms, such as variety, irreducible, etc. Each variety can be associated with a commutative ring, the so-called coordinate ring.
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