Viatcheslav Kharlamov

On the Number of Components of a Complete Intersection of Real Quadrics

Our main results-5pc]Please check the text is ok? concern complete intersections of three real quadrics. We prove that the maximal number B20(N) of connected components that a regular complete intersection of three real quadrics in ℙ N may have differs at most by one from the maximal number of ovals of the submaximal depth ([(N - 1)/2])of a real plane projective curve of degree (d = N + 1). As a consequence, we obtain a lower bound (frac{1} {4}{N}^{2} + O(N))and an upper bound (frac{3} {8}{N}^{2} + O(N))for B20(N).

Welschinger invariants of real del Pezzo surfaces of degree ≥ 2

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of - DKX - 1 real points lying on a connected component of the real part ℝX of X one can trace a real rational curve C ∈ |D|. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov–Witten invariants. Our approach consists in a conversion of Shoval–Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakils extension of the Abramovich–Bertram formula for Gromov–Witten invariants into formulas computing real enumerative invariants.

Topological classification of real Enriques surfaces

We complete the classification of the topological types of real Enriques surfaces started by V. Nikulin. The resulting list contains 87 topological types.

Abundance of Real Lines on Real Projective Hypersurfaces

Our aim is to show that in the case of a generic real hypersurface X of degree 2n−1 in a projective space of dimension n+1 the number NR of real lines onX is not less than approximately the square root of the number NC of complex lines. More precisely, NR ≥ (2n− 1)!!, while due to Don Zagier [GM] NC ∼ √ 27 π (2n− 1) 3 2 , so that logNR ≥ n log 2n+O(n) = 1 2 logNC. Note that NR, unlike NC, depends not only on n but also on the choice of X . The key point of our estimate is an appropriate signed count of the real lines that makes the sum invariant. This sum, which we denote by N e R , is nothing but the Euler number of a suitable vector bundle (see 3.1). Its evaluation gives finally the conclusion:

Welschinger Invariants Revisited

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

Abundance of 3-Planes on Real Projective Hypersurfaces

We show that a generic real projective n-dimensional hypersurface of odd degree d, such that

Qualitative aspects of counting real rational curves on real K3 surfaces

4(n-2)=left( {begin{array}{c}d+3 3end{array}}right)

Welschinger invariants of real Del Pezzo surfaces of degree >- 3

4(n-2)=d+33, contains “many” real 3-planes, namely, in the logarithmic scale their number has the same rate of growth,